450 results on '"Caputo derivative"'
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2. Analysis of impulsive Caputo fractional integro‐differential equations with delay.
- Author
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Zada, Akbar, Riaz, Usman, Jamshed, Junaid, Alam, Mehboob, and Kallekh, Afef
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CAPUTO fractional derivatives , *EQUATIONS , *INTEGRO-differential equations - Abstract
The main focus of this manuscript is to study an impulsive fractional integro‐differential equation with delay and Caputo fractional derivative. The existence solution of such a class of fractional differential equations is discussed for linear and nonlinear case with the help of direct integral method. Moreover, Banach's fixed point theorem and Schaefer's fixed point theorem are use to discuss the uniqueness and at least one solution of the said fractional differential equations, respectively. Some hypothesis and inequalities are utilize to present four different types of Hyers–Ulam stability of the mentioned impulsive integro‐differential equation. Example is provide for the illustration of main results. [ABSTRACT FROM AUTHOR]
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- 2024
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3. A new graph theoretic analytical method for nonlinear distributed order fractional ordinary differential equations by clique polynomial of cocktail party graph.
- Author
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Nirmala, A. N. and Kumbinarasaiah, S.
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FRACTIONAL differential equations ,POLYNOMIALS ,CAPUTO fractional derivatives ,FRACTIONAL calculus ,CONVERGENT evolution - Abstract
In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods. [ABSTRACT FROM AUTHOR]
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- 2024
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4. (α+2)-Dimensional fractional evolution equation: group classification, symmetries, reduction and conservation laws.
- Author
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Hejazi, S. Reza and Naderifard, Azadeh
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CAPUTO fractional derivatives , *FRACTIONAL calculus , *LIE algebras , *CONSERVATION laws (Physics) , *SYMMETRY - Abstract
A preliminary group classification based on symmetry operators is applied to study invariance properties of the time-fractional $(\alpha +2)$(α+2)-dimensional fractional evolution equation. The concepts of Riemann–Liouville and Caputo fractional derivatives are used in this study. The similarity variables obtained from symmetries and one-dimensional optimal systems of constructed Lie algebras are used in order to find the group-invariant solutions of the equation. Finally conservation laws of the equation are derived via a modified version of Noether’s theorem. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Convergence of Runge–Kutta-based convolution quadrature for semilinear fractional differential equations.
- Author
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Zhao, Jingjun, Kong, Jiameng, and Xu, Yang
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FRACTIONAL differential equations , *CAPUTO fractional derivatives - Abstract
For solving the semilinear fractional differential equations with the nonsmooth force term, we construct a class of Runge–Kutta-based convolution quadrature. Moreover, we analyse the convergence of the proposed scheme. In addition, we employ the fast Runge–Kutta approximation to reduce the calculation cost. Finally, we give some numerical experiments to verify the theoretical results. [ABSTRACT FROM AUTHOR]
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- 2024
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6. Mathematical modeling of poliomyelitis virus with vaccination and post‐paralytic syndrome dynamics using Caputo and ABC fractional derivatives.
- Author
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Azroul, Elhoussine and Bouda, Sara
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CAPUTO fractional derivatives , *POLIOMYELITIS vaccines , *BASIC reproduction number , *POLIOVIRUS , *FRACTIONAL differential equations , *GLOBAL analysis (Mathematics) - Abstract
In this study, using Caputo and ABC derivatives, we present a mathematical analysis of two fractional models for poliomyelitis, considering the presence of vaccination (V) and a post‐paralytic class (A). The existence and uniqueness of solutions are proved. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ is computed. Local and global stability of the disease‐free stationary state, depending on the threshold R0$$ {\mathcal{R}}_0 $$, is provided, along with conditions for the existence of an endemic stationary state. Moreover, we performed a sensitivity analysis to study the influence of all biological parameters on R0$$ {\mathcal{R}}_0 $$. We concluded our study with numerical simulations to illustrate the models' dynamics and to compare the trajectories of Caputo and ABC solutions. We found that the Caputo and ABC operators are both convenient for the modelization of the poliomyelitis disease. However, the ABC operator not only refined the Caputo operator by removing singularity from the kernel expression but also brought out heredity and memory in the model's characteristics. [ABSTRACT FROM AUTHOR]
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- 2024
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7. New study on Cauchy problems of fractional stochastic evolution systems on an infinite interval.
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Sivasankar, S., Nadhaprasadh, K., Kumar, M. Sathish, Al‐Omari, Shrideh, and Udhayakumar, R.
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CAPUTO fractional derivatives , *STOCHASTIC analysis , *STOCHASTIC systems , *CAUCHY problem - Abstract
In this study, we examine whether mild solutions to a fractional stochastic evolution system with a fractional Caputo derivative on an infinite interval exist and are attractive. We use semigroup theory, fractional calculus, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. There are several suggested sufficient requirements for the existence of mild solutions to the stated problem. Examples that highlight the key findings are provided. [ABSTRACT FROM AUTHOR]
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- 2024
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8. Fractional‐order modeling of Chikungunya virus transmission dynamics.
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Chavada, Anil, Pathak, Nimisha, and Khirsariya, Sagar R.
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CHIKUNGUNYA virus , *CAPUTO fractional derivatives , *CONTINUOUS time models , *NUMERICAL analysis , *INFECTIOUS disease transmission - Abstract
This article presents two innovative mathematical models for the dynamics of Chikungunya virus contamination by using Caputo fractional derivative. By applying the recently developed numerical technique to find the approximate solutions for the Chikungunya virus system which allowing us for the valuable insights. Through a rigorous analysis of the obtained numerical and graphical solutions, the impact of fractional orders on the infection dynamics is thoroughly examined. Additionally, Banach's fix point theorm is used to investigates the existence, uniqueness, and stability properties of the solutions, providing a deeper understanding of the key parameters that affect the spread and persistence of the infection. [ABSTRACT FROM AUTHOR]
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- 2024
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9. A homotopy-based computational scheme for two-dimensional fractional cable equation.
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Kumar, C. V. Darshan, Prakasha, D. G., Veeresha, P., and Kapoor, Mamta
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CAPUTO fractional derivatives , *FIXED point theory , *FRACTIONAL differential equations , *ELECTROPHYSIOLOGY , *EQUATIONS - Abstract
In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach's fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering. [ABSTRACT FROM AUTHOR]
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- 2024
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10. Dust acoustic nonlinearity of nonlinear mode in plasma to compute temporal and spatial results.
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Khan, Aziz, Sinan, Muhammad, Bibi, Sumera, Shah, Kamal, Hleili, Manel, Abdalla, Bahaaeldin, and Abdeljawad, Thabet
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SOUND pressure ,CAPUTO fractional derivatives ,PLASMA instabilities ,SHOCK waves ,SOUND waves - Abstract
Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution. [ABSTRACT FROM AUTHOR]
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- 2024
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11. Modeling Klebsiella pneumonia infections and antibiotic resistance dynamics with fractional differential equations: insights from real data in North Cyprus.
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Amilo, David, Bagkur, Cemile, and Kaymakamzade, Bilgen
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KLEBSIELLA infections ,KLEBSIELLA pneumoniae ,DRUG resistance in bacteria ,CAPUTO fractional derivatives ,ERTAPENEM - Abstract
This study presents an enhanced fractional-order mathematical model for analyzing the dynamics of Klebsiella pneumonia infections and antibiotic resistance over time. The model incorporates fractional Caputo derivative operators and kernel, to provide a more comprehensive understanding of the complex temporal dynamics. The model consists of three groups: Susceptible (S), Infected (I), and Resistant (R) individuals, each controlled by a fractional differential equation. The model represents the interaction between infection, recovery from infection, and the possible development of antibiotic resistance in susceptible individuals. The existence, uniqueness, stability, and alignment of the model's prediction to the observed data were analyzed and buttressed with numerical simulations. The results show that imipenem has the highest efficacy compared with ertapenem and meropenem category drugs. The estimated reproduction number and reproduction coefficient illustrate the potential impact of this model in improving treatment strategies, while the memory effects highlight the advantages of fractional differentiation. The model predicts an increased possibility of antibiotic resistance despite effective treatment, suggesting a new treatment approach. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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12. Darbo's Fixed-Point Theorem: Establishing Existence and Uniqueness Results for Hybrid Caputo–Hadamard Fractional Sequential Differential Equations.
- Author
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Yaseen, Muhammad, Mumtaz, Sadia, George, Reny, Hussain, Azhar, and Nabwey, Hossam A.
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HYBRID systems , *EXISTENCE theorems , *CAPUTO fractional derivatives , *FIXED point theory , *FRACTIONAL calculus , *FRACTIONAL differential equations , *DIFFERENTIAL equations - Abstract
This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo's fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo's fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo's theorem to analyze the solutions' existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory. [ABSTRACT FROM AUTHOR]
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- 2024
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13. EXPLICIT CRITERIA FOR THE OSCILLATION OF CAPUTO TYPE FRACTIONAL-ORDER DELAY DIFFERENTIAL EQUATIONS.
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DEEPALAKSHMI, RAJASEKAR, SARAVANAN, SIVAGANDHI, THANDAPANI, ETHIRAJU, and TUNC, ERCAN
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DELAY differential equations ,CAPUTO fractional derivatives ,OSCILLATIONS ,FRACTIONAL differential equations - Abstract
New explicit criteria for the oscillation of all solutions of first and second-order delay differential equations including the Caputo fractional derivative are presented. Examples illustrating the importance and novelty of the main results are included. [ABSTRACT FROM AUTHOR]
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- 2024
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14. Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators.
- Author
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Serag, Hassan M., Almoneef, Areej A., El-Badawy, Mahmoud, and Hyder, Abd-Allah
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CAPUTO fractional derivatives , *EULER-Lagrange equations , *DYNAMICAL systems - Abstract
This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system's state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed. [ABSTRACT FROM AUTHOR]
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- 2024
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15. Constructing the fractional series solutions for time-fractional K-dV equation using Laplace residual power series technique.
- Author
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Yadav, Sanjeev, Vats, Ramesh Kumar, and Rao, Anjali
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POWER series , *ELASTIC wave propagation , *CAPUTO fractional derivatives , *FRACTIONAL powers , *DECOMPOSITION method , *HEAT equation , *LAPLACE transformation - Abstract
In this article, we construct the series solution of the time-fractional Korteveg de Vries (K-dV) equation through a computational approach named as Laplace residual power series (LRPS) that combines the Laplace transform with the residual power series method (RPS). Time-fractional K-dV equation is used to modeled various real life phenomena like propagation of waves in elastic rods, dispersion effects in shallow coastal regions, anomalous diffusion observed in financial markets. The Caputo fractional derivative is used in the formulation of time-fractional K-dV equation. LRPS method is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity without any perturbation, discretization and linearization. To assess the effectiveness of proposed computational strategy, we perform a comparative analysis among the fractional residual power series method, the Adomian decomposition method, and the RPS method. Additionally, we examine the convergence of the fractional series solution across different α values and assess the solution's behavior as the time domain increased. The efficiency and authenticity of the LRPS method is shown by computing the absolute error, relative error and residual error. This work is supported by 2D and 3D graphical representations made in accordance with Maple and MATLAB. [ABSTRACT FROM AUTHOR]
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- 2024
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16. APPROXIMATE ANALYTICAL SOLUTION OF GENERALIZED FRACTAL EQUAL-WIDTH WAVE EQUATION.
- Author
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Yun QIAO
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WAVE equation , *NONLINEAR differential equations , *CAPUTO fractional derivatives , *ANALYTICAL solutions , *DECOMPOSITION method , *FRACTALS , *PARTIAL differential equations - Abstract
In this paper, a generalized equal width wave equation involving space fractal derivatives and time Caputo fractional derivatives is studied and its approximate analytical solution is presented by the Adomian decomposition method. An example shows that the method is efficient to solve fractal non-linear partial differential equations. [ABSTRACT FROM AUTHOR]
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- 2024
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17. Semilinear multi-term fractional in time diffusion with memory.
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Vasylyeva, Nataliya, Kochubei, Anatoly, Shepelsky, Dmitry, and Orsingher, Enzo
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FRACTIONAL calculus ,FRACTIONAL differential equations ,SEMILINEAR elliptic equations ,BOUNDARY value problems ,MEMORY ,CAPUTO fractional derivatives ,ELLIPTIC operators - Abstract
In this study, the initial-boundary value problems to semilinear integro-differential equations with multi-term fractional Caputo derivatives are analyzed. A particular case of these equations models oxygen diffusion through capillaries. Under proper requirements on the given data in the model, the classical and strong solvability of these problems for any finite time interval [0, T] are proved via so-called continuation method. The key point in this approach is finding suitable a priori estimates of a solution in the fractional Hölder and Sobolev spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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18. Pontryagin's maximum principle for the Roesser model with a fractional Caputo derivative.
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YUSUBOV, Shakir Sh. and MAHMUDOV, Elimhan N.
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PONTRYAGIN'S minimum principle ,OPTIMAL control theory ,CAPUTO fractional derivatives ,FRACTIONAL integrals ,INTEGRAL equations ,MAXIMUM principles (Mathematics) - Abstract
In this paper, we study the modern mathematical theory of the optimal control problem associated with the fractional Roesser model and described by Caputo partial derivatives, where the functional is given by the Riemann-Liouville fractional integral. In the formulated problem, a new version of the increment method is applied, which uses the concept of an adjoint integral equation. Using the Banach fixed point principle, we prove the existence and uniqueness of a solution to the adjoint problem. Then the necessary and sufficient optimality condition is derived in the form of the Pontryagin's maximum principle. Finally, the result obtained is illustrated by a concrete example. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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19. Efficient Numerical Solutions for Fuzzy Time Fractional Diffusion Equations Using Two Explicit Compact Finite Difference Methods.
- Author
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Batiha, Belal
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FINITE difference method ,HEAT equation ,FRACTIONAL differential equations ,PARTIAL differential equations ,FINITE differences ,CAPUTO fractional derivatives - Abstract
This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev's scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev's is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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20. A shifted Chebyshev operational matrix method for pantograph‐type nonlinear fractional differential equations.
- Author
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Yang, Changqing and Lv, Xiaoguang
- Subjects
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NONLINEAR differential equations , *CAPUTO fractional derivatives , *DIFFERENTIAL equations , *CHEBYSHEV polynomials , *CHEBYSHEV approximation , *FRACTIONAL differential equations - Abstract
In this study, we investigate and analyze an approximation of the Chebyshev polynomials for pantograph‐type fractional‐order differential equations. First, we construct the operational matrices of pantograph and Caputo fractional derivatives using Chebyshev interpolation. Then, the obtained matrices are utilized to approximate the fractional derivative. We also provide a detailed convergence analysis in terms of the weighted square norm. Finally, we describe and discuss the results of three numerical experiments conducted to confirm the applicability and accuracy of the computational scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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21. Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation.
- Author
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Alikhanov, Anatoly A., Asl, Mohammad Shahbazi, and Huang, Chengming
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FRACTIONAL calculus , *CAPUTO fractional derivatives , *WAVE equation , *FRACTIONAL integrals , *EQUATIONS , *NUMERICAL analysis - Abstract
This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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22. Fuzzy Fractional Caputo Derivative of Susceptible-Infectious- Removed Epidemic Model for Childhood Diseases.
- Author
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Subramanian, Suganya, Kumaran, Agilan, Ravichandran, Srilekha, Venugopal, Parthiban, Dhahri, Slim, and Ramasamy, Kavikumar
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CAPUTO fractional derivatives , *JUVENILE diseases , *FRACTIONAL differential equations , *LAPLACE transformation , *EPIDEMICS - Abstract
In this work, the susceptible-infectious-removed (SIR) dynamics are considered in relation to the effects on the health system. With the help of the Caputo derivative fractional-order method, the SIR epidemic model for childhood diseases is designed. Subsequenly, a set of sufficient conditions ensuring the existence and uniqueness of the addressed model by choosing proper fuzzy approximation methods. In particular, the fuzzy Laplace method along with the Adomian decomposition transform were employed to better understand the dynamical structures of childhood diseases. This leads to the development of an efficient methodology for solving fuzzy fractional differential equations using Laplace transforms and their inverses, specifically with the Caputo sense derivative. This innovative approach facilitates the numerical resolution of the problem and numerical simulations are executed for considering parameter values. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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23. On collocation-Galerkin method and fractional B-spline functions for a class of stochastic fractional integro-differential equations.
- Author
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Masti, I. and Sayevand, K.
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FRACTIONAL calculus , *FUNCTIONAL equations , *INTEGRAL equations , *CAPUTO fractional derivatives , *FRACTIONAL integrals , *FRACTIONAL differential equations , *INTEGRO-differential equations - Abstract
In recent years, as detailed in several monographs, derivations of the fractional differential equations and fractional integral equations are based on random functional or stochastic equations, with the output that physical interpretation of the resulting fractional derivatives and fractional integrals has been elusive. In many different sciences and problems such as biological systems, environmental quality and natural resources engineering, and so on, stochastic equations and in some cases random functional have appeared. Despite the widespread use of stochastic fractional integro-differential equations (SFIDE), the analytical solution of this equation is not easy and in some cases it is impossible. Therefore, the existence of an efficient and appropriate numerical method can solve this problem. In this study and based on fractional derivative in the Caputo sense, we investigate a class of SFIDE due to Brownian motion by using fractional B-spline basis functions (FB-spline) and with the help of the collocation-Galerkin method as well as the trapezoidal integral law. In other words, a continuous operator problem (here was named as SFIDE) is transformed into a discrete problem by limited sets of basis functions with common assumptions and approximation methods. In the follow-up a system of linear equations is generated, which makes the analysis of the method be efficient. As an important advantage of this combined method is its flexible and easy implementation. Another advantage of the method is its ability to be implemented for different types of linear, non-linear and system of SFIDE, which are discussed in the body of manuscript. An accurate upper bound is obtained and some theorems are established on the stability and convergence analysis. The computational cost is estimated from the sum of the number arithmetic operations and a lower bound for approximation of this equation is formulated. Finally, by examining several examples, the computational performance of the proposed method effectively verifies the applicability and validity of the suggested scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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24. A fractional-order model for computer viruses and some solution associated with residual power series method.
- Author
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Al-Jarrah, A., Alquran, M., Freihat, A., Al-Omari, S., and Nonlaopon, K.
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COMPUTER viruses , *COMPUTER simulation , *CAPUTO fractional derivatives , *EPIDEMIOLOGICAL models , *RUNGE-Kutta formulas , *POWER series , *SMART devices - Abstract
Awareness of virus spreading is an important issue for building various defence strategies and protecting personal computers, smart devices, network devices, etc. In this research work, we develop epidemiological models to address this problem and introduce certain modified epidemiological fractional SAIR model, where we consider the fractional derivative in Caputo sense. We utilize the residual power series method to construct approximate solutions for the governing system. To show the efficiency and suitability of the proposed technique, we introduce a comparative between the obtained solutions and those solutions that are constructed using the fourth-order Runge-Kutta method. We derive some numerical results by considering specific values for the parameters in the governing model, and then we depict some of these results into two and three dimensions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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25. Existence and Hyers–Ulam stability of solutions to a nonlinear implicit coupled system of fractional order.
- Author
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Zada, Akbar, Ali, Asfandyar, and Riaz, Usman
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CAPUTO fractional derivatives , *DIFFERENTIAL equations , *NONLINEAR systems , *FRACTIONAL differential equations - Abstract
In this typescript, we study system of nonlinear implicit coupled differential equations of arbitrary (non–integer) order having nonlocal boundary conditions on closed interval [0, 1] with Caputo fractional derivative. We establish sufficient conditions for the existence, at least one and a unique solution of the proposed coupled system with the help of Krasnoselskii's fixed point theorem and Banach contraction principle. Moreover, we scrutinize the Hyers–Ulam stability for the considered problem. We present examples to illustrate our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
26. Bifurcation analysis of impulsive fractional-order Beddington-DeAngelis prey-predator model.
- Author
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Alidousti, Javad, Fardi, Mojtaba, and Al-Omari, Shrideh
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BIFURCATION theory ,LYAPUNOV functions ,CAPUTO fractional derivatives ,FUNCTIONAL analysis ,STABILITY constants ,CHAOS theory - Abstract
In this paper, a fractional density-dependent prey-predator model has been considered. Certain reading of local and global stabilities of an equilibrium point of a system was extracted and conducted by applying fractional systems' stability theorems along with Lyapunov functions. Meanwhile, the persistence of the aforementioned system has been discussed and claimed to imply a local asymptotic stability for the given positive equilibrium point. Moreover, the presented model was extended to a periodic impulsive model for the prey population. Such an expansion was implemented through the periodic catching of the prey species and the periodic releasing of the predator population. By studying the effect of changing some of the system's parameters and drawing their bifurcation diagram, it was observed that different periodic solutions appear in the system. However, the effect of an impulse on the system subjects the system to various dynamic changes and makes it experience behaviors including cycles, period-doubling bifurcation, chaos and coexistence as well. Finally, by comparing the fractional system with the classic one, it has been concluded that the fractional system is more stable than its classical one. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
27. High order approximations of solutions to initial value problems for linear fractional integro-differential equations.
- Author
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Ford, Neville J., Pedas, Arvet, and Vikerpuur, Mikk
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INITIAL value problems , *INTEGRO-differential equations , *SINGULAR integrals , *CAPUTO fractional derivatives , *FRACTIONAL differential equations , *LINEAR equations - Abstract
We consider a general class of linear integro-differential equations with Caputo fractional derivatives and weakly singular kernels. First, the underlying initial value problem is reformulated as an integral equation and the possible singular behavior of its exact solution is determined. After that, using a suitable smoothing transformation and spline collocation techniques, the numerical solution of the problem is discussed. Optimal convergence estimates are derived and a superconvergence result of the proposed method is established. The obtained theoretical results are supported by numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
28. Global existence and convergence results for a class of nonlinear time fractional diffusion equation.
- Author
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Huy Tuan, Nguyen
- Subjects
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HEAT equation , *REACTION-diffusion equations , *NAVIER-Stokes equations , *CAPUTO fractional derivatives , *CAUCHY problem , *NONLINEAR equations , *HAMILTON-Jacobi equations , *FRACTIONAL differential equations - Abstract
This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when α → 1 − . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when α → 1 − . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
29. ANALYSIS OF SOCIAL MEDIA ADDICTION MODEL WITH SINGULAR OPERATOR.
- Author
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MOMANI, SHAHER, CHAUHAN, R. P., KUMAR, SUNIL, and HADID, SAMIR
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SOCIAL media addiction , *NONLINEAR functional analysis , *FIXED point theory , *COMPULSIVE behavior , *CAPUTO fractional derivatives , *FUNCTIONAL analysis - Abstract
In this work, we study a dynamic model of social media addiction. Addiction to social media is a behavioral addiction that is very complex in nature. To study such kinds of complex behavior, the fractional-order operators are very useful due to their memory nature. Considering all these facts, we consider the fractional and fractal-fractional operators with a singular kernel to investigate the social media addiction model. This paper's main aim is to demonstrate the importance of non-classical-order derivatives in the investigation of the social media addiction model. First, we propose the model using Caputo fractional derivative and present some basic mathematical computations. The existence condition for a solution of the model system is presented via fixed-point theory. Furthermore, the behavior of the social media addiction model is examined using the fractal-fractional concept with the Caputo operator, which, due to its memory effect, is very effective in modeling. The existence and uniqueness of the solution are investigated using fixed point theory. Ulam–Hyers stability is demonstrated using nonlinear functional analysis. We demonstrate the simulated numerical results graphically for the proposed models via numerical methods based on Lagrange polynomial interpolation. The results are plotted for choices of arbitrary-order parameter values. Based on our findings, we can say the fractal-fractional operators provide more realistic information about the complexity of the dynamics of the social media addiction model. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
30. A New Hybrid Optimal Auxiliary Function Method for Approximate Solutions of Non-Linear Fractional Partial Differential Equations.
- Author
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Ashraf, Rashid, Nawaz, Rashid, Alabdali, Osama, Fewster-Young, Nicholas, Ali, Ali Hasan, Ghanim, Firas, and Alb Lupaş, Alina
- Subjects
- *
FRACTIONAL differential equations , *PARTIAL differential equations , *NONLINEAR differential equations , *CAPUTO fractional derivatives , *NONLINEAR equations , *FRACTIONAL calculus , *HYBRID systems , *POISSON'S equation - Abstract
This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville's fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method's effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
31. The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron.
- Author
-
Verma, Akanksha, Sumelka, Wojciech, and Yadav, Pramod Kumar
- Subjects
- *
DUFFING equations , *FRACTIONAL differential equations , *NONLINEAR differential equations , *CAPUTO fractional derivatives , *MEAN square algorithms , *APPROXIMATION algorithms - Abstract
This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. Analysis and Numerical Computations of the Multi-Dimensional, Time-Fractional Model of Navier-Stokes Equation with a New Integral Transformation.
- Author
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Yuming Chu, Rashid, Saima, Kubra, Khadija Tul, Inc, Mustafa, Zakia Hammouch, and Osman, M. S.
- Subjects
NAVIER-Stokes equations ,NUMERICAL analysis ,INTEGRAL equations ,CAPUTO fractional derivatives ,DECOMPOSITION method - Abstract
The analytical solution of the multi-dimensional, time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transform decomposition method is presented in this article. The aforesaid model is analyzed by employing Caputo fractional derivative. We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods, respectively. The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems. The exact and estimated solutions are delineated via numerical simulation. The proposed analysis indicates that the projected configuration is extremely meticulous, highly efficient, and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. A novel tuberculosis model incorporating a Caputo fractional derivative and treatment effect via the homotopy perturbation method.
- Author
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Olayiwola, Morufu Oyedunsi and Adedokun, Kamilu Adewale
- Subjects
- *
TUBERCULOSIS , *CAPUTO fractional derivatives , *MEDICAL personnel , *MYCOBACTERIUM tuberculosis , *TREATMENT effectiveness , *COMMUNICABLE diseases - Abstract
Background: Tuberculosis (TB), caused by Mycobacterium tuberculosis, is a contagious infectious disease that primarily targets the lungs but can also impact other critical systems such as the bones, joints, and neurological system. Despite significant efforts to combat TB, it remains a major global health concern. To address this challenge, this study aims to explore and evaluate various tuberculosis control approaches using a mathematical modeling framework. Results: The study utilized a novel SEITR mathematical model to investigate the impact of treatment on physical limitations in tuberculosis. The model underwent qualitative analysis to validate key aspects, including positivity, existence, uniqueness, and boundedness. Disease-free and endemic equilibria were identified, and both local and global stability of the model was thoroughly examined using the derived reproduction number. To estimate the impact of each parameter on each compartment, sensitivity analysis was conducted, and numerical simulations were performed using Maple 18 software with the homotopy perturbation method. The obtained results are promising and highlight the potential of the proposed interventions to significantly reduce tuberculosis virus prevalence. The findings emphasize the significance of fractional-order analysis in understanding the effectiveness of treatment strategies for mitigating tuberculosis prevalence. The study suggests that the time fractional dynamics of TB treatment correspond to the treatment's efficacy, as the conceptual results showed that non-local interactions between the disease and the treatment may lead to more accurate ways of eradicating tuberculosis in real-world scenarios. These insights contribute to a better understanding of effective treatment strategies and their potential impact on tuberculosis control and public health. Conclusions: In conclusion, scientists, researchers, and healthcare personnel are urged to take action and utilize the discoveries from this research to facilitate the eradication of the hazardous tuberculosis bacteria. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. Results on the controllability of Caputo’s fractional descriptor systems with constant delays.
- Author
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Sikora, Beata
- Subjects
- *
CONTROLLABILITY in systems engineering , *CAPUTO fractional derivatives , *DESCRIPTOR systems , *ELECTRIC circuits , *EQUATIONS of state , *LINEAR systems - Abstract
The paper investigates the controllability of fractional descriptor linear systems with constant delays in control. The Caputo fractional derivative is considered. Using the Drazin inverse and the Laplace transform, a formula for solving of the matrix state equation is obtained. New criteria of relative controllability for Caputo’s fractional descriptor systems are formulated and proved. Both constrained and unconstrained controls are considered. To emphasize the importance of the theoretical studies, an application to electrical circuits is presented as a practical example. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. A novel numerical technique for solving time fractional nonlinear diffusion equations involving weak singularities.
- Author
-
Ghosh, Bappa and Mohapatra, Jugal
- Subjects
- *
BURGERS' equation , *NONLINEAR equations , *CAPUTO fractional derivatives , *FINITE differences - Abstract
In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time t=0$$ t=0 $$. The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Numerical simulation of SIR childhood diseases model with fractional Adams–Bashforth method.
- Author
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Rahul and Prakash, Amit
- Subjects
- *
JUVENILE diseases , *MEDICAL model , *INTEGRAL calculus , *COMPUTER simulation , *PROGRAMMING languages , *CAPUTO fractional derivatives , *CAUCHY integrals - Abstract
This work investigates a fractional Susceptible Infected Recovered (SIR) model to study childhood disease. We analyzed the proposed model by applying Caputo, Caputo–Fabrizio, and Atangana–Baleanu fractional derivatives. Here, we use the fractional Adams–Bashforth method to solve the childhood disease model with nonlocal operator. The proposed numerical technique is developed by combining the fundamental theorem of integral calculus with Lagrange's interpolation. This numerical approach is more efficient than the other existing numerical techniques and is easily computed using Matlab as a programming language. The existence and uniqueness of this fractional model are studied with the fixed‐point theorem. These fractional derivatives show different asymptotic behaviors for the distinct values of the fractional order. The numerical results are presented graphically as well as in tabulated form for some value of fractional order. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
37. Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function.
- Author
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Rashid, Saima, Kubra, Khadija Tul, and Abualnaja, Khadijah M.
- Subjects
- *
SET functions , *CAPUTO fractional derivatives , *EQUATIONS , *STATISTICS - Abstract
In this article, the Elzaki homotopy perturbation transform method (EHPTM) is profusely employed to discover the approximate solutions of fractional‐order (FO) heat‐like equations. To show this, we first establish the Elzaki transform in the context of the Atangana–Baleanu fractional derivative in the Caputo sense (ABC) and then extend it to heat‐like equations. Our suggested approach has been reinforced by convergence and error analysis. The validity of the novel technique is tested with the aid of some illustrative examples. Comparative analysis has been established for both fractional and integer‐order solutions. EHPTM is considered to be an appropriate and convenient approach for solving FO time‐dependent linear and nonlinear partial differential problems. Plots and tables are being used to reveal the findings. The relatively high validity and reliability of the present approach are also reflected by the comparative solution analysis by means of statistical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
38. New results of global Mittag-Leffler synchronization on Caputo fuzzy delayed inertial neural networks.
- Author
-
Xiangnian Yin, Hongmei Zhang, Hai Zhang, Weiwei Zhang, and Jinde Cao
- Subjects
FUZZY algorithms ,NEURAL circuitry ,SUBSTITUTION (Logic) ,CAPUTO fractional derivatives ,FRACTIONAL calculus - Abstract
This article is devoted to discussing the problem of global Mittag-Leffler synchronization (GMLS) for the Caputo-type fractional-order fuzzy delayed inertial neural networks (FOFINNs). First of all, both inertial and fuzzy terms are taken into account in the system. For the sake of reducing the influence caused by the inertia term, the order reduction is achieved by the measure of variable substitution. The introduction of fuzzy terms can evade fuzziness or uncertainty as much as possible. Subsequently, a nonlinear delayed controller is designed to achieve GMLS. Utilizing the inequality techniques, Lyapunov's direct method for functions and Razumikhin theorem, the criteria for interpreting the GMLS of FOFINNs are established. Particularly, two inferences are presented in two special cases. Additionally, the availability of the acquired results are further confirmed by simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
39. IMPULSIVE DIFFERENTIAL EQUATIONS WITH ERDÉLYI-KOBER BOUNDARY CONDITIONS AND FRACTIONAL DERIVATIVES OF CAPUTO TYPE.
- Author
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Roy, Bandita and Bora, Swaroop Nandan
- Subjects
BOUNDARY value problems ,FRACTIONAL calculus ,CAPUTO fractional derivatives ,FRACTIONAL differential equations ,IMPULSIVE differential equations ,BANACH spaces ,NONLINEAR operators ,INTEGRAL equations - Abstract
This work takes up the study of the existence and uniqueness of solutions to a class of impulsive fractional boundary value problems of order q ∈ (1, 2). The results are obtained by using multiple base points and by transforming the boundary value problem into an equivalent integral equation in a Banach space. Various properties of fractional calculus and a number of familiar fixed point theorems are used to obtain the results. A nonlinear operator is defined in a Banach space whose fixed point gives the solution. The obtained results can be seen as more general since Erdélyi-Kober integrals are known to be more general operators in fractional calculus and they reduce to Riemann-Liouville integrals with a power weight. An example is also provided which illustrates our abstract result. [ABSTRACT FROM AUTHOR]
- Published
- 2023
40. Discrete Approximation of Solutions of the Cauchy Problem for a Linear Homogeneous Differential-Operator Equation with a Caputo Fractional Derivative in a Banach Space.
- Author
-
Kokurin, M. M.
- Subjects
- *
CAPUTO fractional derivatives , *BANACH spaces , *LINEAR differential equations , *CAUCHY problem , *FRACTIONAL differential equations , *HYPERGEOMETRIC functions - Abstract
In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order α ∈ (0, 1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative.
- Author
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Bohner, Martin, Domoshnitsky, Alexander, Padhi, Seshadev, and Srivastava, Satyam Narayan
- Subjects
- *
CAPUTO fractional derivatives , *GREEN'S functions , *DERIVATIVES (Mathematics) , *FUNCTIONAL differential equations , *FRACTIONAL differential equations , *DIFFERENTIAL inequalities - Abstract
In this paper, the functional differential equation (D C a + α x) (t) + ∑ i = 0 m (T i x (i)) (t) = f (t) , t ∈ [ a , b ] , with Caputo fractional derivative D C a + α is studied. The operators Ti act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be presented from the first equation and substituted then to another. For two-point problems with this equation, assertions about negativity of Green's functions and their derivatives with respect to t are obtained. Our technique is based on an analog of the Vallée-Poussin theorem for differential inequalities, which is proven in our paper and gives necessary and sufficient conditions of negativity of Green's functions and their derivatives for two-point problems: there exists a positive function v satisfying corresponding boundary conditions and the inequality (D C a + α ν) (t) + ∑ i = 0 m (T i v (i)) (t) < 0 , t ∈ [ a , b ] . Choosing the function v, we obtain explicit sufficient tests of sign-constancy of Green's functions and its derivatives. It is demonstrated that these tests cannot be improved in a general case. Influences of delays on these sufficient conditions are analyzed. It is demonstrated that the tests can be essentially improved for "small" deviations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
42. Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise.
- Author
-
Egwu, Bernard A. and Yan, Yubin
- Subjects
FINITE element method ,WAVE equation ,CAPUTO fractional derivatives ,NUMERICAL analysis ,SEMILINEAR elliptic equations - Abstract
We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order α ∈ (1 , 2) . The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. SOME RESULTS ON MIXED FRACTIONAL INTEGRODIFFERENTIAL EQUATION IN MATRIX MB-SPACE.
- Author
-
KHARAT, V. V. and RESHIMKAR, ANAND R.
- Subjects
NONLINEAR systems ,CAPUTO fractional derivatives ,STOCHASTIC analysis ,FIXED point theory ,DIFFERENTIAL equations - Abstract
In this article, we study the best approximation of nonlinear mixed fractional integrodifferential equation with Caputo fractional derivative by using a class of stochastic matrix control functions. Next, using the fixed point method, we study the Ulam-Hyers and Ulam-Hyers-Rassias stability of the non-linear fractional integrodifferential equation of the mixed type in MB-space. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
44. Fractional Langevin Coupled System with Stieltjes Integral Conditions.
- Author
-
Majeed, Rafia, Zhang, Binlin, and Alam, Mehboob
- Subjects
- *
STIELTJES integrals , *NONLINEAR functional analysis , *CAPUTO fractional derivatives , *FUNCTIONAL analysis - Abstract
This article outlines the necessary requirements for a coupled system of fractional order boundary value involving the Caputo fractional derivative, including its existence, uniqueness, and various forms of Ulam stability. We demonstrate the existence and uniqueness of the proposed coupled system by using the cone-type Leray–Schauder result and the Banach contraction principle. Based on the traditional method of nonlinear functional analysis, the stability is examined. An example is used to provide a clear illustration of our main results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
45. Optimal Control for Systems Modeled by the Diffusion-Wave Equation.
- Author
-
Postnov, S. S.
- Subjects
- *
CAPUTO fractional derivatives , *EQUATIONS , *WAVE equation , *MOMENTS method (Statistics) - Abstract
The article deals with an optimal control problem for the model system described by the one-dimensional inhomogeneous diffusion-wave equation that is a generalization of the wave equation to the case when the time derivative is replaced with the fractional Caputo derivative. In the general case, we consider both boundary and distributed controls which are Lebesgue -summable functions, with and . We state and study the two types of optimal control problems: The problem of finding a minimal norm control for a given control time and the performance problem of finding a control that brings the system to a given state in the minimal time for a given constraint on the control norm. The study bases on using an exact solution to the diffusion-wave equation, which allows us to reduce the optimal control problem to an infinite-dimensional -moment problem. We also examine the similar finite-dimensional -moment problem that uses an approximate solution to the diffusion-wave equation and analyze the well-posedness and solvability of this problem. Also, we exhibit some example of calculating the boundary control by using the finite-dimensional -moment problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
46. Numerical Solution of Fractional Models of Dispersion Contaminants in the Planetary Boundary Layer.
- Author
-
Koleva, Miglena N. and Vulkov, Lubin G.
- Subjects
- *
ATMOSPHERIC boundary layer , *ADVECTION-diffusion equations , *CAPUTO fractional derivatives , *BOUNDARY layer (Aerodynamics) , *POLLUTANTS , *DISPERSION (Atmospheric chemistry) , *DISPERSION (Chemistry) - Abstract
In this study, a numerical solution for degenerate space–time fractional advection–dispersion equations is proposed to simulate atmospheric dispersion in vertically inhomogeneous planetary boundary layers. The fractional derivative exists in a Caputo sense. We establish the maximum principle and a priori estimates for the solutions. Then, we construct a positivity-preserving finite-difference scheme, using monotone discretization in space and L1 approximation on the non-uniform mesh for the time derivative. We use appropriate grading techniques for the time–space mesh in order to overcome the boundary degeneration and weak singularity of the solution at the initial time. The computational results are demonstrated on the Gaussian fractional model as well on the boundary layers defined by height-dependent wind flow and diffusitivity, especially for the Monin–Obukhov model. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
47. A Numerical Solution of Generalized Caputo Fractional Initial Value Problems.
- Author
-
Saadeh, Rania, A. Abdoon, Mohamed, Qazza, Ahmad, and Berir, Mohammed
- Subjects
- *
INITIAL value problems , *CAPUTO fractional derivatives - Abstract
In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. The presented technique is considered to be an extension to the original Adams–Bashforth–Moulton approach. Numerical simulations and figures are presented and discussed, in order to show the efficiency of the proposed method. In the future, we anticipate that the provided generalized Caputo fractional derivative and the suggested method will be utilized to create and simulate a wide variety of generalized Caputo-type fractional models. We have included examples to demonstrate the accuracy of the present method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
48. Collocation Based Approximations for a Class of Fractional Boundary Value Problems.
- Author
-
Soots, Hanna Britt, Latt, Kaido, and Pedas, Arvet
- Subjects
- *
BOUNDARY value problems , *CAPUTO fractional derivatives , *INTEGRAL equations , *COLLOCATION methods , *INTEGRO-differential equations - Abstract
A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to z = Dα Capy, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
49. Some novel mathematical analysis on the fractional‐order 2019‐nCoV dynamical model.
- Author
-
Owoyemi, Abiodun Ezekiel, Sulaiman, Ibrahim Mohammed, Kumar, Pushpendra, Govindaraj, Venkatesan, and Mamat, Mustafa
- Subjects
- *
SARS-CoV-2 , *MATHEMATICAL analysis , *CAPUTO fractional derivatives , *VIRAL transmission , *COVID-19 - Abstract
Since December 2019, the whole world has been facing the big challenge of Covid‐19 or 2019‐nCoV. Some nations have controlled or are controlling the spread of this virus strongly, but some countries are in big trouble because of their poor control strategies. Nowadays, mathematical models are very effective tools to simulate outbreaks of this virus. In this research, we analyze a fractional‐order model of Covid‐19 in terms of the Caputo fractional derivative. First, we generalize an integer‐order model to a fractional sense, and then, we check the stability of equilibrium points. To check the dynamics of Covid‐19, we plot several graphs on the time scale of daily and monthly cases. The main goal of this content is to show the effectiveness of fractional‐order models as compared to integer‐order dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. A linear Galerkin numerical method for a quasilinear subdiffusion equation.
- Author
-
Płociniczak, Łukasz
- Subjects
- *
GALERKIN methods , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *EQUATIONS , *NONLINEAR functions - Abstract
We couple the L1 discretization for the Caputo derivative in time with the spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove the stability and convergence of the method with spectral accuracy in space. The temporal order depends on the regularity of the solution in time. Furthermore, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result, we find exact asymptotic values of the error constants along with their remainders for discretizations of the Caputo derivative and fractional integrals. These constants are the smallest possible, which improves previously established results from the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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