615 results on '"Caputo derivative"'
Search Results
202. The fractional natural decomposition method: theories and applications.
- Author
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Rawashdeh, Mahmoud S.
- Subjects
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MATHEMATICAL decomposition , *PROBLEM solving , *MATHEMATICAL analysis , *APPROXIMATION theory , *FRACTIONAL calculus , *MATHEMATICAL models - Abstract
In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
203. Fractional-order gradient descent learning of BP neural networks with Caputo derivative.
- Author
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Wang, Jian, Wen, Yanqing, Gou, Yida, Ye, Zhenyun, and Chen, Hua
- Subjects
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ARTIFICIAL neural networks , *BACK propagation , *INFORMATION processing , *ENERGY function , *SIMULATION methods & models - Abstract
Fractional calculus has been found to be a promising area of research for information processing and modeling of some physical systems. In this paper, we propose a fractional gradient descent method for the backpropagation (BP) training of neural networks. In particular, the Caputo derivative is employed to evaluate the fractional-order gradient of the error defined as the traditional quadratic energy function. The monotonicity and weak (strong) convergence of the proposed approach are proved in detail. Two simulations have been implemented to illustrate the performance of presented fractional-order BP algorithm on three small datasets and one large dataset. The numerical simulations effectively verify the theoretical observations of this paper as well. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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204. FRACTIONAL MODELING OF HYPERBOLIC BIOHEAT TRANSFER EQUATION DURING THERMAL THERAPY.
- Author
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KUMAR, P. and RAI, K. N.
- Subjects
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HEAT transfer , *TAYLOR'S series , *TISSUE mechanics , *FRACTIONAL calculus , *LEGENDRE'S functions , *MATHEMATICAL models - Abstract
In this paper, we have developed a fractional hyperbolic bioheat transfer (FHBHT) model by applying fractional Taylor series formula to the single-phase-lag constitutive relation. A new hybrid numerical scheme that combines the multi-resolution and multi-scale computational property of Legendre wavelets based on fractional operational matrix has been used to find the numerical solution of the present problem. This study demonstrates that FHBHT model can provide a unified approach for analyzing heat transfer within living biological tissues, as standard hyperbolic bioheat transfer (SHBHT) and Pennes models are particular cases of FHBHT model. The effect of phase lag time and order of fractional derivative on temperature distribution within living biological tissues for both SHBHT and FHBHT models have been studied and shown graphically. It has been observed that thermal signal propagates more easily with larger values of order of fractional derivative within living biological tissues. The time interval for achieving temperature range of thermal treatment for different models have been studied and compared. It is least for Pennes model, highest for FHBHT model and in between them for SHBHT model. The whole analysis is presented in dimensionless form. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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- View/download PDF
205. Fractional Diffusion in a Solid with Mass Absorption.
- Author
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Povstenko, Yuriy, Kyrylych, Tamara, and Rygał, Grażyna
- Subjects
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RIESZ spaces , *LATTICE theory , *VECTOR spaces , *LAPLACIAN matrices , *GRAPH theory , *INTEGRAL transforms - Abstract
The space-time-fractional diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplacian is considered in the case of axial symmetry. Mass absorption (mass release) is described by a source term proportional to concentration. The integral transform technique is used. Different particular cases of the solution are studied. The numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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206. Stability results for partial fractional differential equations with noninstantaneous impulses.
- Author
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Abbas, Saïd, Benchohra, Mouffak, Alsaedi, Ahmed, and Zhou, Yong
- Subjects
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FRACTIONAL calculus , *DARBOUX transformations , *PARTIAL differential equations , *BANACH spaces , *MATHEMATICAL inequalities , *INTEGRAL equations - Abstract
In this article, we investigate some uniqueness and Ulam's type stability concepts for the Darboux problem of partial functional differential equations with noninstantaneous impulses and delay in Banach spaces. The main techniques rely on fractional calculus, integral equations and inequalities. Two examples are also provided to illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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207. Fractional optimal control problem for differential system with delay argument.
- Author
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Bahaa, G.
- Subjects
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FRACTIONAL calculus , *OPTIMAL control theory , *HILBERT space , *CONSTRAINTS (Physics) , *CAPUTO fractional derivatives - Abstract
In this paper, we apply the classical control theory to a fractional differential system in a bounded domain. The fractional optimal control problem (FOCP) for differential system with time delay is considered. The fractional time derivative is considered in a Riemann-Liouville sense. We first study the existence and the uniqueness of the solution of the fractional differential system with time delay in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a FOCP is considered as a function of both state and control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of a right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in detail. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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208. Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order.
- Author
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Alkan, Sertan and Hatipoglu, Veysel Fuat
- Subjects
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APPROXIMATE solutions (Logic) , *INTEGRO-differential equations , *VOLTERRA equations , *FREDHOLM equations , *CAPUTO fractional derivatives , *FRACTIONAL calculus - Abstract
In this study, sinc-collocation method is introduced for solving Volterra-Fredholm integro-differential equations of fractional order. Fractional derivative is described in the Caputo sense. Obtained results are given to literature as a new theorem. Some numerical examples are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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209. Interpretation of Fractional Derivatives as Reconstruction from Sequence of Integer Derivatives.
- Author
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Tarasov, Vasily E.
- Subjects
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FRACTIONAL calculus , *MATHEMATICAL sequences , *DERIVATIVES (Mathematics) , *SAMPLING theorem , *HADAMARD matrices - Abstract
In this paper, we propose an "informatic" interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders as reconstruction from infinite sequence of standard derivatives of integer orders. The reconstruction is considered with respect to orders of derivatives. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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210. Model of Thin Viscous Fluid Sheet Flow within the Scope of Fractional Calculus: Fractional Derivative with and No Singular Kernel.
- Author
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Atangana, Abdon and Koca, Ilknur
- Subjects
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VISCOUS flow , *FRACTIONAL calculus , *EXISTENCE theorems , *NONLINEAR equations , *FIXED point theory , *COMPUTER simulation - Abstract
A comparative analysis of a model of thin viscous fluid sheet flow between Caputo and Caputo-Fabrizio derivative with fractional order was performed in this work. Sides-by-sides we presented some properties of both derivatives, and then we examined the existence of the exact solution of both nonlinear equations via the fixed-point theorem. A detailed study of the uniqueness of analysis for both models is presented. Numerical simulations are presented to access the difference between both models. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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211. Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method.
- Author
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Wang, Yanxin and Zhu, Li
- Subjects
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NUMERICAL solutions to integro-differential equations , *FRACTIONAL calculus , *APPROXIMATION theory , *WAVELETS (Mathematics) , *NONLINEAR systems , *MATHEMATICAL singularities - Abstract
In this paper, a wavelet numerical method for solving nonlinear Volterra integro-differential equations of fractional order is presented. The method is based upon Euler wavelet approximations. The Euler wavelet is first presented and an operational matrix of fractional-order integration is derived. By using the operational matrix, the nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which is solved through known numerical algorithms. Also, various types of solutions, with smooth, non-smooth, and even singular behavior have been considered. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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212. Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses.
- Author
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Povstenko, Y.
- Subjects
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CAPUTO fractional derivatives , *FRACTIONAL calculus , *ELECTRICAL harmonics , *HEAT conduction , *THERMAL stresses - Abstract
Time-nonlocal generalization of the classical Fourier law with the “long-tail” power kernel can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and leads to the time-fractional heat conduction equation with the Caputo derivative. Fractional heat conduction equation with the harmonic source term under zero initial conditions is studied. Different formulations of the problem for the standard parabolic heat conduction equation and for the hyperbolic wave equation appearing in thermoelasticity without energy dissipation are discussed. The integral transform technique is used. The corresponding thermal stresses are found using the displacement potential. [ABSTRACT FROM PUBLISHER]
- Published
- 2016
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213. Numerical approach for solving space fractional order diffusion equations using shifted Chebyshev polynomials of the fourth kind.
- Author
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SWEILAM, Nasser Hassan, NAGY, Abdelhameed Mohamed, and Elaziz El-SAYED, Adel Abd
- Subjects
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CHEBYSHEV polynomials , *FRACTIONAL calculus , *HEAT equation , *FINITE difference method , *APPROXIMATION theory - Abstract
In this paper, a new approach for solving space fractional order diffusion equations is proposed. The fractional derivative in this problem is in the Caputo sense. This approach is based on shifted Chebyshev polynomials of the fourth kind with the collocation method. The finite difference method is used to reduce the equations obtained by our approach for a system of algebraic equations that can be efficiently solved. Numerical results obtained with our approach are presented and compared with the results obtained by other numerical methods. The numerical results show the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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214. A high order numerical scheme for variable order fractional ordinary differential equation.
- Author
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Cao, Jianxiong and Qiu, Yanan
- Subjects
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NUMERICAL analysis , *FRACTIONAL calculus , *ORDINARY differential equations , *STOCHASTIC convergence , *DERIVATIVES (Mathematics) - Abstract
In this paper, we derive a high order numerical scheme for variable order fractional ordinary differential equation by establishing a second order numerical approximation to variable order Riemann–Liouville fractional derivative. The scheme is strictly proved to be stable and convergent with second order accuracy, which is higher than some recently derived schemes. Finally, some numerical examples are presented to demonstrate the theoretical analysis and verify the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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215. A novel method for solving second order fractional eigenvalue problems.
- Author
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Reutskiy, S.Yu.
- Subjects
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FRACTIONAL calculus , *EIGENVALUES , *PROBLEM solving , *FRACTIONAL differential equations , *ANALYTICAL solutions , *COLLOCATION methods - Abstract
The paper presents a new numerical method for solving eigenvalue problems for fractional differential equations. It combines two techniques: the method of external excitation (MEE) and the backward substitution method (BSM). The first one is a mathematical model of physical measurements when a mechanical, electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies. The BSM consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. The numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
216. Theory and applications of a more general form for fractional power series expansion.
- Author
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Jaradat, I., Al-Dolat, M., Al-Zoubi, K., and Alquran, M.
- Subjects
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FRACTIONAL calculus , *POWER series , *CAPUTO fractional derivatives , *ITERATIVE methods (Mathematics) , *DIFFERENTIAL equations - Abstract
The latent potentialities and applications of fractional calculus present a mathematical challenge to establish its theoretical framework. One of these challenges is to have a compact and self-contained fractional power series representation that has a wider application scope and allows studying analytical properties. In this letter, we introduce a new more general form of fractional power series expansion, based on the Caputo sense of fractional derivative, with corresponding convergence property. In order to show the functionality of the proposed expansion, we apply the corresponding iterative fractional power series scheme to solve several fractional (integro-)differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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217. Simulation and numerical solution of fractional order Ebola virus model with novel technique
- Author
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Ali Raza, Aqeel Ahmad, Muhammad Sajid Iqbal, Muhammad Farman, and Ali Akgül
- Subjects
education.field_of_study ,lcsh:Medical technology ,Steady state ,Ebola virus ,Laplace transform ,lcsh:Biotechnology ,Population ,lcsh:TP155-156 ,caputo derivative ,medicine.disease_cause ,Fractional calculus ,Nonlinear system ,Operator (computer programming) ,sensitivity analysis ,lcsh:R855-855.5 ,lcsh:TP248.13-248.65 ,medicine ,fractional-order ebola model ,Applied mathematics ,Uniqueness ,lcsh:Chemical engineering ,education ,existence and uniqueness ,Mathematics - Abstract
In this paper, nonlinear fractional order Ebola virus mathematical model is discussed for the complex transmission of the epidemic problems. We developed the fractional order Ebola virus transmission model for the treatment and control to reduce its effect on a population which play an important role for public health. Qualitative analysis has been made to verify the the steady state and uniqueness of the system is also developed for reliable results. Caputo fractional derivative operator φ i ∈ (0,1] works to achieve the fractional differential equations. Laplace with Adomian Decomposition Method successfully solved the fractional differential equations. Ultimately, numerical simulations are also developed to evaluate the effects of the device parameter on spread of disease and effect of fractional parameter φ i on obtained solution which can also be assessed by tabulated results.
- Published
- 2020
218. Nonlinear multi-term fractional differential equations with Riemann-Stieltjes integro-multipoint boundary conditions
- Author
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Ymnah Alruwaily, Ahmed Alsaedi, Bashir Ahmad, and Sotiris K. Ntouyas
- Subjects
General Mathematics ,lcsh:Mathematics ,Fixed-point theorem ,Riemann–Stieltjes integral ,Function (mathematics) ,caputo derivative ,Fixed point ,lcsh:QA1-939 ,Fractional calculus ,riemann-stieltjes integral ,Nonlinear system ,fixed point ,existence of solutions ,Applied mathematics ,Boundary value problem ,Uniqueness ,multipoint boundary conditions ,Mathematics - Abstract
In this paper, we consider a nonlinear multi-term Caputo fractional differential equation with nonlinearity depending on the unknown function together with its lower-order Caputo fractional derivatives and equipped with Riemann-Stieltjes integro multipoint boundary conditions. The given problem is transformed to an equivalent fixed point problem, which is then solved with the aid of standard fixed point theorems to establish the existence and uniqueness results for the problem at hand. Examples are constructed for the illustration of the obtained results.
- Published
- 2020
219. A class of impulsive vibration equation with fractional derivatives
- Author
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Mei Jia, Xue Wang, and Xiping Liu
- Subjects
Class (set theory) ,lcsh:Mathematics ,General Mathematics ,fractional derivative ,impulsive vibration equation ,Multiplicity (mathematics) ,caputo derivative ,lcsh:QA1-939 ,Nonlinear boundary conditions ,Fractional calculus ,Vibration ,Monotone polygon ,nonlinear boundary conditions ,Order (group theory) ,Applied mathematics ,Mathematics - Abstract
In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives. By using monotone iterative technique, some new results on the multiplicity for solutions of the equations under nonlinear boundary conditions are obtained, and the properties of the solutions are discussed. Finally, the practicability of our results is discussed through a concrete example.
- Published
- 2020
220. Numerical Solution of Nonlinear Fractional Boundary Value Problems.
- Author
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Pedas, Arvet and Tamme, Enn
- Subjects
- *
NONLINEAR analysis , *FRACTIONAL calculus , *NUMERICAL solutions to boundary value problems , *ALGORITHMS , *STOCHASTIC convergence - Abstract
The numerical solution of boundary value problems for nonlinear fractional differential equations is discussed. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties and spline collocation techniques, the attainable order of convergence of the proposed algorithms is studied, theoretically and numerically. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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221. On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives
- Author
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Briceyda B. Delgado and Jorge Eduardo Macías-Díaz
- Subjects
Statistics and Probability ,Curl (mathematics) ,QA299.6-433 ,Vector operator ,Generalization ,Riemann–Liouville derivative ,fractional vector operators ,Statistical and Nonlinear Physics ,Space (mathematics) ,Dirac operator ,Caputo derivative ,Fractional calculus ,Divergence ,symbols.namesake ,Helmholtz decomposition theorem ,fractional div-curl systems ,symbols ,QA1-939 ,Applied mathematics ,Thermodynamics ,QC310.15-319 ,Helmholtz decomposition ,Mathematics ,Analysis - Abstract
In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.
- Published
- 2021
222. Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions
- Author
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Adem Kilicman, Ahcene Merad, and Saadoune Brahimi
- Subjects
fractional derivatives ,General Mathematics ,Finite difference method ,A priori estimate ,fractional advection–diffusion equation ,Caputo derivative ,Numerical integration ,Fractional calculus ,finite difference schemes ,integral conditions ,Range (mathematics) ,Operator (computer programming) ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,Boundary value problem ,Uniqueness ,Engineering (miscellaneous) ,Mathematics - Abstract
In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types ∫01vx,tdx and ∫01xnvx,tdx. The existence and uniqueness of the given problem’s solution is proved using the method of the energy inequalities known as the “a priori estimate” method relying on the range density of the operator generated by the considered problem. The approximate solution for this problem with these new kinds of boundary conditions is established by using a combination of the finite difference method and the numerical integration. Finally, we give some numerical tests to illustrate the usefulness of the obtained results.
- Published
- 2021
- Full Text
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223. Effects of Fractional Derivatives with Different Orders in SIS Epidemic Models
- Author
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Anna Chiara Lai, Paola Loreti, Caterina Balzotti, and Mirko D'Ovidio
- Subjects
General Computer Science ,Applied Mathematics ,Susceptible–Infected–Susceptible (SIS) models ,Context (language use) ,Total population ,QA75.5-76.95 ,Birth–death process ,Caputo derivative ,Theoretical Computer Science ,Fractional calculus ,Operator (computer programming) ,Modeling and Simulation ,Electronic computers. Computer science ,Caputo–Fabrizio operator ,susceptible–infected–susceptible (SIS) models ,Applied mathematics ,Quantitative Biology::Populations and Evolution ,Mathematics - Abstract
We study epidemic Susceptible–Infected–Susceptible (SIS) models in the fractional setting. The novelty is to consider models in which the susceptible and infected populations evolve according to different fractional orders. We study a model based on the Caputo derivative, for which we establish existence results of the solutions. Furthermore, we investigate a model based on the Caputo–Fabrizio operator, for which we provide existence of solutions and a study of the equilibria. Both models can be framed in the context of SIS models with time-varying total population, in which the competition between birth and death rates is macroscopically described by the fractional orders of the derivatives. Numerical simulations for both models and a direct numerical comparison are also provided.
- Published
- 2021
224. CONTROLLABILITY CRITERIA FOR TIME-DELAY FRACTIONAL SYSTEMS WITH A RETARDED STATE.
- Author
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SIKORA, BEATA
- Subjects
TIME delay systems ,FRACTIONAL calculus ,MATHEMATICAL formulas ,NUMERICAL analysis ,CONTROL theory (Engineering) - Abstract
The paper is concerned with time-delay linear fractional systems with multiple delays in the state. A formula for the solution of the discussed systems is presented and derived using the Laplace transform. Definitions of relative controllability with and without constraints for linear fractional systems with delays in the state are formulated. Relative controllability, both with and without constraints imposed on control values, is discussed. Various types of necessary and sufficient conditions for relative controllability and relative U-controllability are established and proved. Numerical examples illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
225. Numerical treatment of a well-posed Chebyshev Tau method for Bagley-Torvik equation with high-order of accuracy.
- Author
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Mokhtary, P.
- Subjects
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MATHEMATICAL regularization , *CHEBYSHEV polynomials , *FRACTIONAL calculus , *MICROELECTROMECHANICAL systems , *GAMMA functions - Abstract
The main purpose of this study is to develop and analyze a new high-order operational Tau method based on the Chebyshev polynomials as basis functions for obtaining the numerical solution of Bagley-Torvik equation which has a important role in the fractional calculus. It is shown that some derivatives of the solutions of these equations have a singularity at origin. To overcome this drawback we first change the original equation into a new equation with a better regularity properties by applying a regularization process and thereby the operational Chebyshev Tau method can be applied conveniently. Our proposed method has two main advantages. First, the algebraic form of the Tau discretization of the problem has an upper triangular structure which can be solved by forward substitution method. Second, Tau approximation of the problem converges to the exact ones with a highly rate of convergence under a more general regularity assumptions on the input data in spite of the singularity behavior of the exact solution. Numerical results are presented which confirm the theoretical results obtained and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
226. Effects of the fractional order and magnetic field on the blood flow in cylindrical domains.
- Author
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Ali Shah, Nehad, Vieru, Dumitru, and Fetecau, Constantin
- Subjects
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MAGNETIC fields , *FRACTIONAL powers , *MAGNETOHYDRODYNAMICS , *MAGNETIC particles , *MATHEMATICAL models , *FRACTIONAL calculus , *BLOOD flow , *REYNOLDS number - Abstract
In this paper, based on the magnetohydrodynamics approach, the blood flow along with magnetic particles through a circular cylinder is studied. The fluid is acted by an oscillating pressure gradient and an external magnetic field. The study is based on a mathematical model with Caputo fractional derivatives. The model of ordinary fluid, corresponding to time-derivatives of integer order, is obtained as a particular case. Closed forms of the fluid velocity and magnetic particles velocity are obtained by means of the Laplace and finite Hankel transforms. Effects of the order of Caputo's time-fractional derivatives and of the external magnetic field on flow parameters of both blood and magnetic particles are studied. Numerical simulations and graphical illustrations are used in order to study the influence of the fractional parameter α , Reynolds number and Hartmann number on the fluid and particles velocity. The results highlights that, models with fractional derivatives bring significant differences compared to the ordinary model. This fact can be an important advantage for some practical problems. It also results that the blood velocity, as well as that of magnetic particles, is reduced under influence of the exterior magnetic field. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
227. Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method.
- Author
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Sakar, Mehmet Giyas, Uludag, Fatih, and Erdogan, Fevzi
- Subjects
- *
FRACTIONAL calculus , *NUMERICAL solutions to partial differential equations , *DELAY differential equations , *HOMOTOPY theory , *PERTURBATION theory , *CAPUTO fractional derivatives - Abstract
In this paper, homotopy perturbation method (HPM) is applied to solve fractional partial differential equations (PDEs) with proportional delay in t and shrinking in x . The method do not require linearization or small perturbation. The fractional derivatives are taken in the Caputo sense. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of α are presented graphically. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
228. Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line.
- Author
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Hashemi, M.S. and Baleanu, D.
- Subjects
- *
CAPUTO fractional derivatives , *FRACTIONAL calculus , *GEOMETRIC function theory , *DEPENDENT variables , *ANALYTIC functions - Abstract
We propose a simple and accurate numerical scheme for solving the time fractional telegraph (TFT) equation within Caputo type fractional derivative. A fictitious coordinate ϑ is imposed onto the problem in order to transform the dependent variable u ( x , t ) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of method of line and group preserving scheme (GPS) is proposed to find the approximate solutions. This method preserves the geometric structure of the problem. Power and accuracy of this method has been illustrated through some examples of TFT equation. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
229. Asymptotic Solutions and Circuit Implementations of a Rayleigh Oscillator Including Cubic Fractional Damping Terms.
- Author
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Xiao, Min, Jiang, Guoping, and Cao, Jinde
- Subjects
- *
ELECTRIC circuits , *RAYLEIGH model , *ELECTRIC oscillators , *ASYMPTOTIC distribution , *FRACTIONAL calculus - Abstract
This paper proposes a fractional-order Rayleigh oscillator model, which involves a cubic damping term described by fractional derivatives. The presence of such fractional damping term makes the analysis more difficult. A two-scale expansion method is employed for asymptotic solutions of the fractional-order Rayleigh oscillator. Then, an example is provided to compare the asymptotic solutions with the numerical solutions. The numerical results demonstrate the validity and applicability of the proposed method to solve fractional differential equations with high order fractional terms. Furthermore, an electronic circuit is designed to realize the fractional-order Rayleigh oscillator. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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230. Existence of the Mild Solution for Impulsive Neutral Stochastic Fractional Integro-Differential Inclusions with Nonlocal Conditions.
- Author
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Chadha, Alka and Pandey, Dwijendra
- Abstract
This paper mainly concerns with the existence of a mild solution for impulsive neutral integro-differential inclusions with nonlocal conditions in a separable Hilbert space. Utilizing fixed point theorem for multi-valued operators due to Dhage, we establish the existence result with resolvent operator and η-norm. An illustrative example is provided to show the effectiveness of the established results. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
231. Hopf lemma for the fractional diffusion operator and its application to a fractional free-boundary problem.
- Author
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Roscani, Sabrina D.
- Subjects
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H-spaces , *FRACTIONAL calculus , *OPERATOR theory , *BOUNDARY value problems , *MONOTONE operators - Abstract
We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order α ∈ ( 0 , 1 ) is taken in the Caputo sense. A generalization of the Hopf lemma is proved and then used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
232. Fractional electromagnetic waves in conducting media.
- Author
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Gómez-Aguilar, J.F., Yépez-Martínez, H., Calderón-Ramón, C., Benavidez-Cruz, M., and Morales-Mendoza, L.J.
- Subjects
- *
FRACTIONAL calculus , *ELECTROMAGNETIC waves , *ELECTRICAL conductors , *WAVE equation , *MAXWELL equations , *CURRENT density (Electromagnetism) - Abstract
We present the fractional wave equation in a conducting material. We used a Maxwell’s equations with the assumptions that the charge densityand current densityJwere zero, and that the permeabilityand permittivitywere constants. The fractional wave equation will be examined separately; with fractional spatial derivative and fractional temporal derivative, finally, consider a Dirichlet conditions, the Fourier method was used to find the full solution of the fractional equation in analytic way. Two auxiliary parametersandare introduced; these parameters characterize consistently the existence of the fractional space-time derivatives into the fractional wave equation. A physical relation between these parameters is reported. The fractional derivative of Caputo type is considered and the corresponding solutions are given in terms of the Mittag-Leffler function show fractal space-time geometry different from the classical integer-order model. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
233. Nonlocal electrical diffusion equation.
- Author
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Gómez-Aguilar, J. F., Escobar-Jiménez, R. F., Olivares-Peregrino, V. H., Benavides-Cruz, M., and Calderón-Ramón, C.
- Subjects
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HEAT equation , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *ELECTROCHEMISTRY , *SEMICONDUCTORS , *ELECTRIC fields - Abstract
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is and for the time domain is . We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
234. Mild solution for impulsive neutral fractional partial differential inclusions with nonlocal conditions.
- Author
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Chadha, Alka and Pandey, Dwijendra
- Abstract
In the present paper, we study the existence of a mild solution of a fractional order nonlocal differential inclusion with impulsive condition in a Banach space E. We obtain the sufficient condition for the existence of the mild solution by using a fixed point theorem for multi-valued operators due to Dhage and resolvent semigroup theory with approximate techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
235. An investigation with Hermite Wavelets for accurate solution of Fractional Jaulent–Miodek equation associated with energy-dependent Schrödinger potential.
- Author
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Gupta, A.K. and Ray, S. Saha
- Subjects
- *
WAVELETS (Mathematics) , *FRACTIONAL calculus , *SCHRODINGER equation , *MATRICES (Mathematics) , *APPROXIMATION theory , *HOMOTOPY theory , *FRACTIONAL differential equations , *NONLINEAR theories - Abstract
In the present paper, a wavelet method based on the Hermite wavelet expansion along with operational matrices of fractional derivative and integration is proposed for finding the numerical solution to a coupled system of nonlinear time-fractional Jaulent–Miodek (JM) equations. Consequently, the approximate solutions of fractional Jaulent–Miodek equations acquired by using Hermite wavelet technique were compared with those derived by using optimal homotopy asymptotic method (OHAM) and exact solutions. The present proposed numerical technique is easy, expedient and powerful in computing the numerical solution of coupled system of nonlinear fractional differential equations like Jaulent–Miodek equations. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
236. VARIATIONAL FORMULATION OF PROBLEMS INVOLVING FRACTIONAL ORDER DIFFERENTIAL OPERATORS.
- Author
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JIN, BANGTI, LAZAROV, RAYTCHO, PASCIAK, JOSEPH, and RUNDELL, WILLIAM
- Subjects
- *
DIFFERENTIAL operators , *FRACTIONAL calculus , *BOUNDARY value problems , *NUMERICAL analysis , *EIGENVALUES - Abstract
In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order α ∈ (1, 2) on the unit interval (0, 1). These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space H0α/2(0, 1) but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
237. An Efficient Mechanism to Solve Fractional Differential Equations Using Fractional Decomposition Method
- Author
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Mahmoud S. Alrawashdeh, Ioannis K. Argyros, and Seba A. Migdady
- Subjects
Work (thermodynamics) ,Physics and Astronomy (miscellaneous) ,Relation (database) ,General Mathematics ,natural transform ,02 engineering and technology ,Derivative ,01 natural sciences ,0202 electrical engineering, electronic engineering, information engineering ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics ,Laplace transform ,fractional differential equations ,system of differential equations ,caputo derivative ,Fractional calculus ,010101 applied mathematics ,Chemistry (miscellaneous) ,Mechanism (philosophy) ,020201 artificial intelligence & image processing ,Decomposition method (constraint satisfaction) - Abstract
We present some new results that deal with the fractional decomposition method (FDM). This method is suitable to handle fractional calculus applications. We also explore exact and approximate solutions to fractional differential equations. The Caputo derivative is used because it allows traditional initial and boundary conditions to be included in the formulation of the problem. This is of great significance for large-scale problems. The study outlines the significant features of the FDM. The relation between the natural transform and Laplace transform is a symmetrical one. Our work can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.
- Published
- 2021
238. Mathematical structure of mosaic disease using microbial biostimulants via Caputo and Atangana–Baleanu derivatives
- Author
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Vedat Suat Erturk, Pushpendra Kumar, and Hassan Almusawa
- Subjects
010302 applied physics ,Mathematical model ,Physics ,QC1-999 ,Mosaic disease ,General Physics and Astronomy ,Fixed-point theorem ,02 engineering and technology ,Type (model theory) ,021001 nanoscience & nanotechnology ,Optimal control ,01 natural sciences ,Runge–Kutta method ,Caputo derivative ,Fractional calculus ,Microbial biostimulants ,Kernel (statistics) ,0103 physical sciences ,Applied mathematics ,Fractional mathematical model ,Uniqueness ,Mathematical structure ,Atangana–Baleanu derivative ,0210 nano-technology ,Mathematics - Abstract
In this research collection, we analysed two different fractional non-linear mathematical models of a well-known mosaic epidemic of plants, which is underlying by begomoviruses and is distributed to plants by whitefly. We included the role of natural microbial biostimulants which are used to increase plant performance and protects them against mosaic infection. Cause of the big expansion of the mosaic epidemic in various geographical areas, and its large privative economic and societal impacts, it is of major consequence to define dominant optimal control means of this disease. In this paper, we used Caputo (singular type kernel) and Atangana–Baleanu (Mittag-Leffler type kernel) fractional derivatives to define the structure of the proposed mosaic model. We performed some important existence and uniqueness analyses for both models by the applications of fixed point theory and the Picard–Lindelof technique. We derived the numerical solution of the Caputo fractional model by the application of the fourth-order Runge–Kutta method and the Atangana–Baleanu model by the Predictor–Corrector algorithm. A long-term discussion on the graphical interpretations of both models with different infection transmission rate and application proportion rate of MBs (microbial biostimulants) at different fractional-order values have established. We exemplified that under the case of the Mittag-Leffler kernel, the effects of different fractional-order values are much clear as compared to the singular type kernel. The main contribution of this paper is to study the dynamics of mosaic disease at different transmission rates and MBs application rates in the sense of two different kernel types.
- Published
- 2021
239. Analytical Solutions of the Fractional Mathematical Model for the Concentration of Tumor Cells for Constant Killing Rate
- Author
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Farman Ali, F. D. Zaman, Dumitru Vieru, Najma Ahmed, and Nehad Ali Shah
- Subjects
Laplace transform ,Mathematical model ,Transcendental equation ,General Mathematics ,integral transform ,010102 general mathematics ,Mathematical analysis ,cancer mathematical model ,caputo derivative ,Integral transform ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,analytical solutions ,symbols.namesake ,Computer Science (miscellaneous) ,symbols ,QA1-939 ,Cylindrical coordinate system ,0101 mathematics ,Constant (mathematics) ,Engineering (miscellaneous) ,Bessel function ,Mathematics - Abstract
Two generalized mathematical models with memory for the concentration of tumor cells have been analytically studied using the cylindrical coordinate and the integral transform methods. The generalization consists of the formulating of two mathematical models with Caputo-time fractional derivative, models that are suitable to highlight the influence of the history of tumor evolution on the present behavior of the concentration of cancer cells. The time-oscillating concentration of cancer cells has been considered on the boundary of the domain. Analytical solutions of the fractional differential equations of the mathematical models have been determined using the Laplace transform with respect to the time variable and the finite Hankel transform with respect to the radial coordinate. The positive roots of the transcendental equation with Bessel function J0(r)=0, which are needed in our study, have been determined with the subroutine rn=root(J0(r),r,(2n−1)π/4,(2n+3)π/4),n=1,2,… of the Mathcad 15 software. It is found that the memory effects are stronger at small values of the time, t. This aspect is highlighted in the graphical illustrations that analyze the behavior of the concentration of tumor cells. Additionally, the concentration of cancer cells is symmetric with respect to radial angle, and its values tend to be zero for large values of the time, t.
- Published
- 2021
240. Analytic and numerical solutions of discrete Bagley–Torvik equation
- Author
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M. Motawi Khashan, Gnanaprakasam Britto Antony Xavier, Fahd Jarad, Thabet Abdeljawad, and M. Meganathan
- Subjects
Pure mathematics ,Algebra and Number Theory ,Partial differential equation ,Functional analysis ,Laplace transform ,Applied Mathematics ,010102 general mathematics ,Fractional calculus ,010103 numerical & computational mathematics ,01 natural sciences ,Caputo derivative ,Ordinary differential equation ,QA1-939 ,Research article ,Nabla symbol ,0101 mathematics ,Difference operator ,Bagley–Torvik equation ,Mathematics ,Analysis ,Real number - Abstract
In this research article, a discrete version of the fractional Bagley–Torvik equation is proposed:$$ \nabla _{h}^{2} u(t)+A{}^{C} \nabla _{h}^{\nu }u(t)+Bu(t)=f(t),\quad t>0, $$∇h2u(t)+AC∇hνu(t)+Bu(t)=f(t),t>0,where$00<ν<1or$11<ν<2, subject to$u(0)=a$u(0)=aand$\nabla _{h} u(0)=b$∇hu(0)=b, withaandbbeing real numbers. The solutions are obtained by employing the nabla discrete Laplace transform. These solutions are expressed in terms of Mittag-Leffler functions with three parameters. These solutions are handled numerically for some examples with specific values of some parameters.
- Published
- 2021
241. The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case.
- Author
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Hu, Ye, Li, Changpin, and Li, Hefeng
- Subjects
- *
FINITE difference method , *CAPUTO fractional derivatives , *STOCHASTIC convergence , *LAPLACIAN operator , *FRACTIONAL calculus - Abstract
In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L 1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli–Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
242. On disappearance of chaos in fractional systems.
- Author
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Deshpande, Amey S. and Daftardar-Gejji, Varsha
- Subjects
- *
FRACTIONAL calculus , *DYNAMICAL systems , *CHAOS theory , *POINCARE series , *LORENZ equations - Abstract
In a seminal paper, Grigorenko and Grigorenko [15], numerically studied fractional order dynamical systems (FODS) of the form D α i x i = f i ( x 1 , x 2 , x 3 ) , 0 ≤ α i ≤ 1 , ( i = 1 , 2 , 3 ) ; and showed the existence of chaos in case of fractional Lorenz system when Σ = α 1 + α 2 + α 3 ≤ 3 . Since then voluminous numerical work has been done to explore various FODS, in this regard. It is now an established fact that Σ acts as a chaos controlling parameter. In the present article we take a survey of present literature on chaotic behavior of fractional order dynamical systems. Further we numerically explore fractional Chen, Rössler, Bhalekar–Gejji, Lorenz and Liu systems and observe that chaos always disappear if Σ ≤ 2. The existing examples in the literature along with the systems that we have analyzed lead us to conjecture non-existence of chaos if Σ ≤ 2; which in some sense is a generalization of classical Poincaré–Bendixon theorem for FODS. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
243. Analysis of a class of boundary value problems depending on left and right Caputo fractional derivatives.
- Author
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Antunes, Pedro R.S. and Ferreira, Rui A.C.
- Subjects
- *
BOUNDARY value problems , *CAPUTO fractional derivatives , *MATHEMATICAL regularization , *PROBLEM solving , *FRACTIONAL calculus - Abstract
In this work we study boundary value problems associated to a nonlinear fractional ordinary differential equation involving left and right Caputo derivatives. We discuss the regularity of the solutions of such problems and, in particular, give precise necessary conditions so that the solutions are C 1 ([0, 1]). Taking into account our analytical results, we address the numerical solution of those problems by the a u g m e n t e d -RBF method. Several examples illustrate the good performance of the numerical method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
244. Mathematical modelling of fractional order circuit elements and bioimpedance applications.
- Author
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Moreles, Miguel Angel and Lainez, Rafael
- Subjects
- *
CIRCUIT elements , *RIEMANN hypothesis , *LAPLACIAN matrices , *KIRCHHOFF'S theory of diffraction , *FRACTIONAL calculus - Abstract
In this work a classical derivation of fractional order circuits models is presented. Generalised constitutive equations in terms of fractional Riemann–Liouville derivatives are introduced in the Maxwell’s equations for each circuit element. Next the Kirchhoff voltage law is applied in a RCL circuit configuration. It is shown that from basic properties of Fractional Calculus, a fractional differential equation model with Caputo derivatives is obtained. Thus standard initial conditions apply. Finally, models for bioimpedance are revisited. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
245. On the oscillation of fractional differential equations via \(ψ\)-Hilfer fractional derivative
- Author
-
D. Vivek, E. M. Elsayed, and Kuppusamy Kanagarajan
- Subjects
ψ -hilfer fractional derivative ,Oscillation ,lcsh:T ,Mathematical analysis ,riemann-liouville operator ,lcsh:Q ,caputo derivative ,Fractional differential ,oscillation ,lcsh:Science ,lcsh:Technology ,Fractional calculus ,Mathematics - Abstract
In this paper, we study the oscillatory theory for fractional differential equations (FDEs) via \(ψ\)-Hilfer fractional derivative. Sufficient conditions are established for the oscillation of solutions FDEs.
- Published
- 2019
246. The dynamics of Zika virus with Caputo fractional derivative
- Author
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Muhammad Farhan, Muhammad Altaf Khan, and Saif Ullah
- Subjects
Lyapunov function ,biology ,General Mathematics ,lcsh:Mathematics ,Dynamics (mechanics) ,Zika virus model ,Fractional model ,Model parameters ,numerical results ,biology.organism_classification ,stability analysis ,lcsh:QA1-939 ,Stability (probability) ,Caputo derivative ,Fractional calculus ,Zika virus ,symbols.namesake ,generalized mean value theorem ,symbols ,Applied mathematics ,Mathematics - Abstract
In the present paper, we investigate a fractional model in Caputo sense to explore the dynamics of the Zika virus. The basic results of the fractional Zika model are presented. The local and global stability analysis of the proposed model is obtained when the basic reproduction reproduction number is less or greater than 1. To show the global stability of the fractional Zika model, we use the Lyapunov function theory in fractional environment. Further, we simulate the fractional Zika model to present the graphical results for different values of fractional order and model parameters.
- Published
- 2019
247. Fractional Diffusion in a Solid with Mass Absorption
- Author
-
Yuriy Povstenko, Tamara Kyrylych, and Grażyna Rygał
- Subjects
fractional calculus ,Caputo derivative ,Riesz derivative ,Mittag-Leffler function ,Laplace transform ,Science ,Astrophysics ,QB460-466 ,Physics ,QC1-999 - Abstract
The space-time-fractional diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplacian is considered in the case of axial symmetry. Mass absorption (mass release) is described by a source term proportional to concentration. The integral transform technique is used. Different particular cases of the solution are studied. The numerical results are illustrated graphically.
- Published
- 2017
- Full Text
- View/download PDF
248. A new mathematical formulation for a phase change problem with a memory flux
- Author
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Julieta Bollati, S. Roscani, and Domingo A. Tarzia
- Subjects
RIEMANN–LIOUVILLE DERIVATIVE ,Matemáticas ,EQUIVALENT INTEGRAL RELATION ,General Mathematics ,FRACTIONAL DIFFUSION EQUATION ,FOS: Physical sciences ,General Physics and Astronomy ,Boundary (topology) ,35R35, 26A33, 35C05, 33E20, 80A22 ,01 natural sciences ,purl.org/becyt/ford/1 [https] ,Phase change ,Mathematics - Analysis of PDEs ,Flux (metallurgy) ,Integral relation ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,CAPUTO DERIVATIVE ,MEMORY FLUX ,Physics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,purl.org/becyt/ford/1.1 [https] ,Stefan problem ,Matemática Aplicada ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Sense (electronics) ,Fractional calculus ,010101 applied mathematics ,STEFAN PROBLEM ,CIENCIAS NATURALES Y EXACTAS ,Analysis of PDEs (math.AP) - Abstract
A mathematical model for a one-phase change problem (particularly a Stefan problem) with a memory flux, is obtained. The hypothesis that the weighted sum of fluxes back in time is proportional to the gradient of temperature is considered. The model obtained involves fractional derivatives with respect on time in the sense of Caputo and in the sense of Riemann--Liouville. An integral relationship for the free boundary which is equivalent to the `fractional Stefan condition' is also obtained., 26 pages, 2 figures
- Published
- 2018
249. On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems
- Author
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Nguyen Hoang Luc, Ho Duy Binh, Nguyen Huu Can, and Erdal Karapınar
- Subjects
Work (thermodynamics) ,Algebra and Number Theory ,Partial differential equation ,lcsh:Mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Function (mathematics) ,Derivative ,Nonlinear fractional pseudo-parabolic equation systems ,lcsh:QA1-939 ,01 natural sciences ,Caputo derivative ,Fractional calculus ,Regularity ,010101 applied mathematics ,Nonlinear system ,Initial value problem ,Ordinary differential equation ,Applied mathematics ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this work, we study an initial value problem for a system of nonlinear parabolic pseudo equations with Caputo fractional derivative. Here, we discuss the continuity which is related to a fractional order derivative. To overcome some of the difficulties of this problem, we need to evaluate the relevant quantities of the Mittag-Leffler function by constants independent of the derivative order. Moreover, we present an example to illustrate the theory.
- Published
- 2021
250. Solution method for the time-fractional hyperbolic heat equation
- Author
-
Ioannis Dassios, Francesc Font, Universitat Politècnica de Catalunya. Departament de Mecànica de Fluids, and Universitat Politècnica de Catalunya. GReCEF- Grup de Recerca en Ciència i Enginyeria de Fluids
- Subjects
General Mathematics ,Heat equation ,Mathematical analysis ,Initial conditions ,General Engineering ,Fractional calculus ,Matemàtiques i estadística::Matemàtica aplicada a les ciències [Àrees temàtiques de la UPC] ,Caputo derivative ,Matemàtica aplicada ,Boundary value problem ,Hyperbolic heat equation ,Mathematics - Abstract
In this article, we propose a method to solve the time-fractional hyperbolic heat equation. We first formulate a boundary value problem for the standard hyperbolic heat equation in a finite domain and provide an analytical solution by means of separation of variables and Fourier series. Then, we consider the same boundary value problem for the fractional hyperbolic heat equation. The fractional problem is solved using three different definitions of the fractional derivative: the Caputo fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio and the Atangana–Baleanu. A closed form of the solution is provided for each case. Finally, we compare the solutions of the fractional and the standard problem and show numerically that the solution of the standard hyperbolic heat equation can be retrieved from the solution of the fractional equation in the limit ¿¿2, where ¿ represents the exponent of the fractional derivative
- Published
- 2021
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