28 results
Search Results
2. Error estimates of variational discretization for semilinear parabolic optimal control problems
- Author
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Zuliang Lu, Xuejiao Chen, Chunjuan Hou, and Fei Huang
- Subjects
Discretization ,General Mathematics ,lcsh:Mathematics ,Type (model theory) ,semilinear parabolic equations ,Residual ,Optimal control ,lcsh:QA1-939 ,Backward Euler method ,Omega ,Finite element method ,error estimates ,optimal control problems ,A priori and a posteriori ,Applied mathematics ,finite element methods ,Mathematics - Abstract
In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.
- Published
- 2021
3. Two new preconditioners for mean curvature-based image deblurring problem
- Author
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Rashad Ahmed, Adel M. Al-Mahdi, and Shahbaz Ahmad
- Subjects
Deblurring ,Discretization ,numerical analysis ,Computer science ,General Mathematics ,Numerical analysis ,mean curvature ,Krylov subspace ,ill-posed problem ,image deblurring ,Nonlinear system ,Fixed-point iteration ,preconditioning ,Computer Science::Computer Vision and Pattern Recognition ,Convergence (routing) ,QA1-939 ,Applied mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.
- Published
- 2021
4. A new algorithm based on compressed Legendre polynomials for solving boundary value problems
- Author
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Yingzhen Lin, Hui Zhu, and Liangcai Mei
- Subjects
compressed legendre polynomials ,boundary value problems ,General Mathematics ,error estimation ,QA1-939 ,Applied mathematics ,Boundary value problem ,stability analysis ,Legendre polynomials ,Mathematics ,convergence analysis - Abstract
In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.
- Published
- 2022
5. Elastic transformation method for solving ordinary differential equations with variable coefficients
- Author
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Xiaoxu Dong, Shunchu Li, Pengshe Zheng, and Jing Luo
- Subjects
Transformation (function) ,General Mathematics ,Ordinary differential equation ,variable coefficient ,general solution ,elastic transformation method ,QA1-939 ,Applied mathematics ,laguerre equation ,ordinary differential equation ,Mathematics ,Variable (mathematics) - Abstract
Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.
- Published
- 2022
6. On ψ-Hilfer generalized proportional fractional operators
- Author
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Subhash Alha, Ali Akgül, Idris Ahmed, Fahd Jarad, and Ishfaq Ahmad Mallah
- Subjects
General Mathematics ,weighed space ,QA1-939 ,Applied mathematics ,generalized proportional fractional derivative ,hilfer fractional derivative ,fixed point theorems ,Mathematics ,existence and uniqueness - Abstract
In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.
- Published
- 2022
7. New Lyapunov-type inequalities for fractional multi-point boundary value problems involving Hilfer-Katugampola fractional derivative
- Author
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Wei Zhang, Jifeng Zhang, and Jinbo Ni
- Subjects
Lyapunov function ,General Mathematics ,hilfer-katugampola fractional derivative ,Type (model theory) ,Fractional calculus ,symbols.namesake ,multi-point boundary condition ,symbols ,QA1-939 ,Applied mathematics ,Boundary value problem ,lyapunov-type inequality ,Multi point ,Mathematics - Abstract
In this paper, we present new Lyapunov-type inequalities for Hilfer-Katugampola fractional differential equations. We first give some unique properties of the Hilfer-Katugampola fractional derivative, and then by using these new properties we convert the multi-point boundary value problems of Hilfer-Katugampola fractional differential equations into the equivalent integral equations with corresponding Green's functions, respectively. Finally, we make use of the Banach's contraction principle to derive the desired results, and give a series of corollaries to show that the current results extend and enrich the previous results in the literature.
- Published
- 2022
8. On a boundary value problem for fractional Hahn integro-difference equations with four-point fractional integral boundary conditions
- Author
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Sotiris K. Ntouyas, Thanin Sitthiwirattham, and Varaporn Wattanakejorn
- Subjects
Mathematics::Functional Analysis ,boundary value problems ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,existence ,Fixed-point theorem ,fractional hahn difference ,Fixed point ,Quantum number ,Nonlinear system ,Operator (computer programming) ,QA1-939 ,Applied mathematics ,fractional hahn integral ,Point (geometry) ,Boundary value problem ,Uniqueness ,Mathematics - Abstract
In this paper, we study a boundary value problem consisting of Hahn integro-difference equation supplemented with four-point fractional Hahn integral boundary conditions. The novelty of this problem lies in the fact that it contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. Firstly, we convert the given nonlinear problem into a fixed point problem, by considering a linear variant of the problem at hand. Once the fixed point operator is available, we make use the classical Banach's and Schauder's fixed point theorems to establish existence and uniqueness results. An example is also constructed to illustrate the main results. Several properties of fractional Hahn integral that will be used in our study are also discussed.
- Published
- 2022
9. Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses
- Author
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Kaihong Zhao and Shuang Ma
- Subjects
Class (set theory) ,Mathematics::Functional Analysis ,General Mathematics ,Stability (learning theory) ,stability ,hadamard fractional integral bvp ,contraction mapping principle ,Nonlinear system ,Hadamard transform ,QA1-939 ,Applied mathematics ,Boundary value problem ,Mathematics ,existence and uniqueness - Abstract
This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.
- Published
- 2022
10. The extended Weibull–Fréchet distribution: properties, inference, and applications in medicine and engineering
- Author
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Ekramy A. Hussein, Ahmed Z. Afify, and Hassan M. Aljohani
- Subjects
Distribution (number theory) ,engineering data ,General Mathematics ,Inference ,Failure rate ,Probability density function ,maximum product of spacing estimators ,cramér–von mises estimation ,Frequentist inference ,fréchet distribution ,Generalized extreme value distribution ,QA1-939 ,Fréchet distribution ,Applied mathematics ,simulations ,Mathematics ,Weibull distribution ,extreme value distribution - Abstract
In this paper, a flexible version of the Fréchet distribution called the extended Weibull–Fréchet (EWFr) distribution is proposed. Its failure rate has a decreasing shape, an increasing shape, and an upside-down bathtub shape. Its density function can be a symmetric shape, an asymmetric shape, a reversed-J shape and J shape. Some mathematical properties of the EWFr distribution are explored. The EWFr parameters are estimated using several frequentist estimation approaches. The performance of these methods is addressed using detailed simulations. Furthermore, the best approach for estimating the EWFr parameters is determined based on partial and overall ranks. Finally, the performance of the EWFr distribution is studied using two real-life datasets from the medicine and engineering sciences. The EWFr distribution provides a superior fit over other competing Fréchet distributions such as the exponentiated-Fréchet, beta-Fréchet, Lomax–Fréchet, and Kumaraswamy Marshall–Olkin Fréchet.
- Published
- 2022
11. An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions
- Author
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Nur Nadiah Abd Hamid, Fouad Mohammad Salama, Norhashidah Hj. Mohd. Ali, and Umair Ali
- Subjects
fractional diffusion equation ,Group (mathematics) ,Iterative method ,General Mathematics ,caputo fractional derivative ,Space (mathematics) ,Fractional diffusion ,QA1-939 ,Applied mathematics ,laplace transform ,stability and convergence ,grouping strategy ,finite difference scheme ,Mathematics - Abstract
In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.
- Published
- 2022
12. On stochastic accelerated gradient with non-strongly convexity
- Author
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Xingxing Zha, Yongquan Zhang, Dongyin Wang, and Yiyuan Cheng
- Subjects
least-square regression ,General Mathematics ,Carry (arithmetic) ,logistic regression ,Supervised learning ,Regular polygon ,Lipschitz continuity ,Stochastic approximation ,accelerated stochastic approximation ,Convexity ,Stochastic programming ,convergence rate ,Rate of convergence ,QA1-939 ,Applied mathematics ,Mathematics - Abstract
In this paper, we consider stochastic approximation algorithms for least-square and logistic regression with no strong-convexity assumption on the convex loss functions. We develop two algorithms with varied step-size motivated by the accelerated gradient algorithm which is initiated for convex stochastic programming. We analyse the developed algorithms that achieve a rate of $ O(1/n^{2}) $ where $ n $ is the number of samples, which is tighter than the best convergence rate $ O(1/n) $ achieved so far on non-strongly-convex stochastic approximation with constant-step-size, for classic supervised learning problems. Our analysis is based on a non-asymptotic analysis of the empirical risk (in expectation) with less assumptions that existing analysis results. It does not require the finite-dimensionality assumption and the Lipschitz condition. We carry out controlled experiments on synthetic and some standard machine learning data sets. Empirical results justify our theoretical analysis and show a faster convergence rate than existing other methods.
- Published
- 2022
13. New iterative approach for the solutions of fractional order inhomogeneous partial differential equations
- Author
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Rashid Nawaz, Sumbal Ahsan, Kottakkaran Sooppy Nisar, Dumitru Baleanu, and Laiq Zada
- Subjects
Partial differential equation ,Laplace transform ,Iterative method ,General Mathematics ,lcsh:Mathematics ,fractional order inhomogeneous system ,Interval (mathematics) ,fractional calculus ,lcsh:QA1-939 ,approximate solutions ,Fractional calculus ,Transformation (function) ,Integer ,fractional order roseau-hyman equation ,Applied mathematics ,Decomposition method (constraint satisfaction) ,new iterative method ,Mathematics - Abstract
In this paper, the study of fractional order partial differential equations is made by using the reliable algorithm of the new iterative method (NIM). The fractional derivatives are considered in the Caputo sense whose order belongs to the closed interval [0, 1]. The proposed method is directly extended to study the fractional-order Roseau-Hyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. The obtained results are compared with those obtained by Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Laplace Variational Iteration Method (LVIM) and the Laplace Adominan Decomposition Method (LADM). The results obtained by NIM, show higher accuracy than HPM, LVIM and LADM. The accuracy of the proposed method improves by taking more iterations.
- Published
- 2021
14. Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings
- Author
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J. F. Tang, X. R. Wang, M. Liu, S. S. Chang, and Salahuddin
- Subjects
residual gap function ,General Mathematics ,lcsh:Mathematics ,Hausdorff space ,Solution set ,Inverse ,hausdorff lipschitz continuity ,Monotonic function ,Function (mathematics) ,error bounds ,Lipschitz continuity ,Residual ,relaxed monotonicity ,lcsh:QA1-939 ,generalized f-projection operator ,regularized gap function ,Variational inequality ,Applied mathematics ,generalized vector inverse quasi-variational inequality problems ,global gap function ,bi-mapping ,Mathematics ,strong monotonicity - Abstract
The goal of this paper is further to study a kind of generalized vector inverse quasi-variational inequality problems and to obtain error bounds in terms of the residual gap function, the regularized gap function, and the global gap function by utilizing the relaxed monotonicity and Hausdorff Lipschitz continuity. These error bounds provide effective estimated distances between an arbitrary feasible point and the solution set of generalized vector inverse quasi-variational inequality problems.
- Published
- 2021
15. The stationary distribution of a stochastic rumor spreading model
- Author
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Dapeng Gao, Peng Guo, and Chaodong Chen
- Subjects
Lyapunov function ,Stationary distribution ,Stochastic modelling ,General Mathematics ,lcsh:Mathematics ,White noise ,Rumor ,lcsh:QA1-939 ,stationary distribution ,symbols.namesake ,rumor spreading ,symbols ,threshold ,Applied mathematics ,Ergodic theory ,Uniqueness ,Persistence (discontinuity) ,Mathematics - Abstract
In this paper, we develop a rumor spreading model by introducing white noise into the model. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. Finally, we provide some numerical simulations to illustrate the analytical results.
- Published
- 2021
16. Finite element approximation of time fractional optimal control problem with integral state constraint
- Author
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Jie Liu and Zhaojie Zhou
- Subjects
Discretization ,General Mathematics ,lcsh:Mathematics ,a priori error estimate ,space time finite element method ,Optimal control ,lcsh:QA1-939 ,integral state constraint ,Finite element method ,Piecewise linear function ,Scheme (mathematics) ,Piecewise ,A priori and a posteriori ,Applied mathematics ,time fractional optimal control problem ,Constant (mathematics) ,Mathematics - Abstract
In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.
- Published
- 2021
17. A relaxed generalized Newton iteration method for generalized absolute value equations
- Author
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Senlai Zhu, Yang Cao, and Shi Quan
- Subjects
Generalized Jacobian ,Iterative method ,General Mathematics ,lcsh:Mathematics ,Positive-definite matrix ,globally convergence ,lcsh:QA1-939 ,symbols.namesake ,generalized absolute value equations ,relaxation ,Fixed-point iteration ,Absolute value equation ,symbols ,newton method ,Applied mathematics ,Well-defined ,Coefficient matrix ,Newton's method ,Mathematics - Abstract
To avoid singular generalized Jacobian matrix and further accelerate the convergence of the generalized Newton (GN) iteration method for solving generalized absolute value equations Ax - B|x| = b, in this paper we propose a new relaxed generalized Newton (RGN) iteration method by introducing a relaxation iteration parameter. The new RGN iteration method involves the well-known GN iteration method and the Picard iteration method as special cases. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions. In addition, a specific sufficient condition is studied when the coefficient matrix A is symmetric positive definite. Finally, two numerical experiments arising from the linear complementarity problems are used to illustrate the effectiveness of the new RGN iteration method.
- Published
- 2021
18. A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response
- Author
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Anwar Zeb, Ranjit Kumar Upadhyay, A. Pratap, and Yougang Wang
- Subjects
Lyapunov function ,Hopf bifurcation ,delay ,General Mathematics ,Addiction ,media_common.quotation_subject ,lcsh:Mathematics ,Functional response ,periodic solution ,stability ,lcsh:QA1-939 ,Two stages ,Critical point (mathematics) ,Synthetic drugs ,symbols.namesake ,symbols ,Applied mathematics ,synthetic drugs model ,hopf bifurcation ,Bifurcation ,media_common ,Mathematics - Abstract
This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.
- Published
- 2021
19. Stability of general pathogen dynamic models with two types of infectious transmission with immune impairment
- Author
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B. S. Alofi and S. A. Azoz
- Subjects
Lyapunov function ,pathogen infection ,Steady state (electronics) ,General Mathematics ,lcsh:Mathematics ,cell-to-cell transmission ,lcsh:QA1-939 ,Stability (probability) ,global stability ,Quantitative Biology::Cell Behavior ,symbols.namesake ,immune impairment ,Transmission (telecommunications) ,Exponential stability ,Stability theory ,Bounded function ,symbols ,Applied mathematics ,Quantitative Biology::Populations and Evolution ,Basic reproduction number ,Mathematics - Abstract
In this paper, we investigate the global properties of two general models of pathogen infection with immune deficiency. Both pathogen-to-cell and cell-to-cell transmissions are considered. Latently infected cells are included in the second model. We show that the solutions are nonnegative and bounded. Lyapunov functions are organized to prove the global asymptotic stability for uninfected and infected steady states of the models. Analytical expressions for the basic reproduction number $\mathcal{R}_{0}$ and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. We prove that if $\mathcal{R}_{0}$ < 1 then the uninfected steady state is globally asymptotically stable (GAS), and if $\mathcal{R}_{0}$ > 1 then the infected steady state is GAS. Numerical simulations are performed and used to support the analytical results.
- Published
- 2021
20. On the nonstandard numerical discretization of SIR epidemic model with a saturated incidence rate and vaccination
- Author
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Isnani Darti and Agus Suryanto
- Subjects
Lyapunov function ,Discretization ,Continuous modelling ,General Mathematics ,lcsh:Mathematics ,Finite difference ,dynamically-consistent discretization ,Function (mathematics) ,Nonstandard finite difference scheme ,saturated incidence rate ,local and global stability analysis ,lcsh:QA1-939 ,Euler method ,symbols.namesake ,symbols ,Applied mathematics ,sir epidemic model ,Epidemic model ,lyapunov function ,Mathematics - Abstract
Recently, Hoang and Egbelowo (Boletin de la Sociedad Matematica Mexicana, 2020) proposed a nonstandard finite difference scheme (NSFD) to get a discrete SIR epidemic model with saturated incidence rate and constant vaccination. The discrete model was derived by discretizing the right-hand sides of the system locally and the first order derivative is approximated by the generalized forward difference method but with a restrictive denominator function. Their analysis showed that the NSFD scheme is dynamically-consistent only for relatively small time-step sizes. In this paper, we propose and analyze an alternative NSFD scheme by applying nonlocal approximation and choosing the denominator function such that the proposed scheme preserves the boundedness of solutions. It is verified that the proposed discrete model is dynamically-consistent with the corresponding continuous model for all time-step size. The analytical results have been confirmed by some numerical simulations. We also show numerically that the proposed NSFD scheme is superior to the Euler method and the NSFD method proposed by Hoang and Egbelowo (2020).
- Published
- 2021
21. A basic study of a fractional integral operator with extended Mittag-Leffler kernel
- Author
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Iyad Suwan, Asad Ali, Thabet Abdeljawad, Kottakkaran Sooppy Nisar, Muhammad Samraiz, and Gauhar Rahman
- Subjects
fractional integral ,Mathematics::Complex Variables ,General Mathematics ,Operator (physics) ,Mathematics::Classical Analysis and ODEs ,Function (mathematics) ,Extension (predicate logic) ,Type (model theory) ,symbols.namesake ,Mathematics::Probability ,mittag-leffler function ,Kernel (statistics) ,Mittag-Leffler function ,prabhakar fractional integral ,symbols ,QA1-939 ,Applied mathematics ,Differential (mathematics) ,Mathematics - Abstract
In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.
- Published
- 2021
22. Maximal and minimal iterative positive solutions for p-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term
- Author
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Lishan Liu, Ying Wang, and Limin Guo
- Subjects
General Mathematics ,Function (mathematics) ,Derivative ,Term (time) ,hadamard fractional differential equation ,Nonlinear system ,infinite-point ,Hadamard transform ,positive solution ,p-Laplacian ,QA1-939 ,Applied mathematics ,Boundary value problem ,Fractional differential ,Mathematics ,iterative positive solution - Abstract
In this paper, the maximal and minimal iterative positive solutions are investigated for a singular Hadamard fractional differential equation boundary value problem with a boundary condition involving values at infinite number of points. Green's function is deduced and some properties of Green's function are given. Based upon these properties, iterative schemes are established for approximating the maximal and minimal positive solutions.
- Published
- 2021
23. Modelling chaotic dynamical attractor with fractal-fractional differential operators
- Author
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Youssef El-Khatib and Sonal Jain
- Subjects
Computer science ,General Mathematics ,Chaotic ,Differential operator ,Dynamical system ,Convolution ,Mathematical Operators ,fractal-fractional integral operator ,Fractal ,Attractor ,chaotic attractors ,QA1-939 ,Applied mathematics ,as strange attractor ,Differential (mathematics) ,fractal-fractional differential operators ,Mathematics - Abstract
Differential operators based on convolution have been recognized as powerful mathematical operators able to depict and capture chaotic behaviors, especially those that are not able to be depicted using classical differential and integral operators. While these differential operators have being applied with great success in many fields of science, especially in the case of dynamical system, we have to confess that they were not able depict some chaotic behaviors, especially those with additionally similar patterns. To solve this issue new class of differential and integral operators were proposed and applied in few problems. In this paper, we aim to depict chaotic behavior using the newly defined differential and integral operators with fractional order and fractal dimension. Additionally we introduced a new chaotic operators with strange attractors. Several simulations have been conducted and illustrations of the results are provided to show the efficiency of the models.
- Published
- 2021
24. A high order numerical method for solving Caputo nonlinear fractional ordinary differential equations
- Author
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Xumei Zhang and Junying Cao
- Subjects
higher order numerical scheme ,General Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,caputo derivative ,nonlinear fractional ordinary differential equations ,convergence analysis ,Nonlinear system ,Ordinary differential equation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,QA1-939 ,Order (group theory) ,Applied mathematics ,finite difference method ,Mathematics - Abstract
In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.
- Published
- 2021
25. Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method
- Author
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Yanpeng Gong, Fajie Wang, and Chao Wang
- Subjects
Helmholtz equation ,General Mathematics ,local knot method ,Inverse ,Basis function ,heat conduction ,Boundary knot method ,System of linear equations ,Thermal conduction ,Functionally graded material ,Nonlinear system ,QA1-939 ,Applied mathematics ,semi-analytical meshless method ,nonlinear functionally graded material ,Mathematics ,kirchhoff transformation - Abstract
This paper proposes a local semi-analytical meshless method for simulating heat conduction in nonlinear functionally graded materials. The governing equation of heat conduction problem in nonlinear functionally graded material is first transformed to an anisotropic modified Helmholtz equation by using the Kirchhoff transformation. Then, the local knot method (LKM) is employed to approximate the solution of the transformed equation. After that, the solution of the original nonlinear equation can be obtained by the inverse Kirchhoff transformation. The LKM is a recently proposed meshless approach. As a local semi-analytical meshless approach, it uses the non-singular general solution as the basis function and has the merits of simplicity, high accuracy, and easy-to-program. Compared with the traditional boundary knot method, the present scheme avoids an ill-conditioned system of equations, and is more suitable for large-scale simulations associated with complicated structures. Three benchmark numerical examples are provided to confirm the accuracy and validity of the proposed approach.
- Published
- 2021
26. Oscillation theorems of solution of second-order neutral differential equations
- Author
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Omar Bazighifan, Hammad Alotaibi, Ali Muhib, and Kamsing Nonlaopon
- Subjects
Class (set theory) ,Oscillation ,Differential equation ,General Mathematics ,second-order neutral differential equation ,QA1-939 ,Applied mathematics ,Order (group theory) ,neutral differential equation ,oscillation criteria ,Neutral differential equations ,Mathematics - Abstract
In this paper, we aim to explore the oscillation of solutions for a class of second-order neutral functional differential equations. We propose new criteria to ensure that all obtained solutions are oscillatory. The obtained results can be used to develop and provide theoretical support for and further develop the oscillation study for a class of second-order neutral differential equations. Finally, an illustrated example is given to demonstrate the effectiveness of our new criteria.
- Published
- 2021
27. On solvability of some p-Laplacian boundary value problems with Caputo fractional derivative
- Author
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Dexin Chen and Xiaoping Li
- Subjects
General Mathematics ,boundary value problem ,caputo fractional derivative ,p-Laplacian ,QA1-939 ,Fixed-point theorem ,Applied mathematics ,fixed point theorem ,Boundary value problem ,solvability ,Mathematics ,Fractional calculus - Abstract
The solvability of some $ p $-Laplace boundary value problems with Caputo fractional derivative are discussed. By using the fixed-point theory and analysis techniques, some existence results of one or three non-negative solutions are obtained. Two examples showed that the conditions used in this paper are somewhat easy to check.
- Published
- 2021
28. Fixed point results of an implicit iterative scheme for fractal generations
- Author
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Muhammad Tanveer, Yi-Xia Li, Qingxiu Peng, Haixia Zhang, and Nehad Ali Shah
- Subjects
jungck-ishikawa iteration ,Fractal ,General Mathematics ,Scheme (mathematics) ,fixed point theory ,fractals ,QA1-939 ,Applied mathematics ,Fixed point ,Mathematics - Abstract
In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.
- Published
- 2021
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