Time Fractional Partial Differential Equations Research Articles

Van der Corput lemmas for Mittag-Leffler functions. II. α–directions

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

This paper is concerned with the investigation of series solutions of some time fractional nonlinear systems of partial differential equations. The symmetries and reductions of the discussed time fractional nonlinear systems are presented in a recent work (J. Math. Phys. 57 (2016) 101504). But, their explicit solutions are investigated in the present study by using power series expansion technique. Moreover, the reported solutions are interpreted graphically to highlight the importance of present study.

A semi-analytical method for 1D, 2D and 3D time fractional second order dual-phase-lag model of the heat transfer

In this paper we present a new numerical technique for 1D, 2D, and 3D time-fractional second order dual-phase-lag model of heat transfer. The problem is formulated as a solution of a time fractional partial differential equation (TFPDE) subjected to specific initial and boundary conditions. In the framework of the presented technique we first find an analytical function which satisfies the boundary condition of the original problem at each time moment. Then, using this function we transform the original problem into the problem with zero boundary conditions. This problem admits of the approximate solution in the form of the Fourier series expansion. Applying the Fourier expansion, the original TFPDE is reduced into a sequence of independent multi-term fractional ordinary differential equations (FODEs). These FODEs are solved separately, using the efficient backward substitution method and the Mu¨ntz polynomials in the approximate solution of each FODE. The accuracy and convergence of the proposed method are examined experimentally with several examples. In particular, the equations with coefficients that correspond to the real thermophysical parameters of the biological tissue have been considered. In the last example we demonstrate the ability of the method presented to solve 3D FPDEs with time-dependent coefficients.

Finite Difference Method with Metaheuristic Orientation for Exploration of Time Fractional Partial Differential Equations

Finite Difference Method with Metaheuristic Orientation for Exploration of Time Fractional Partial Differential Equations

Numerical solution of the nonlinear conformable space–time fractional partial differential equations

Numerical solution of the nonlinear conformable space–time fractional partial differential equations

Computational approach based on wavelets for financial mathematical model governed by distributed order fractional differential equation

In this study, for the first time, the approximate solution of Black–Scholes option pricing distributed order time-fractional partial differential equation by means of Legendre and Chebyshev wavelets is considered. The operational matrices of Legendre and Chebyshev wavelets for integer order derivative and distributed order fractional derivative are derived. Furthermore, the combination of Gauss–Legendre quadrature formula and standard Tau method along with the obtained operational matrices reduces the distributed order time-fractional Black–Scholes model (DOTFBSM) into the system of linear algebraic equations. Convergence analysis, error bounds and numerical stability of the proposed approach are discussed in detail. The presented scheme is applied on three test examples and numerical experiments confirm the theoretical results and illustrate robustness of the presented method. The results produced by current approach are found to be more accurate than some available results.

A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

Comparison between solutions of a two-dimensional time-fractional diffusion-reaction equation through Lie symmetries

In this paper, exact and numerical solutions of two dimensional time-fractional diffusion-reaction equation involving the Riemann-Liouville derivative are determined, by applying a procedure that combines the Lie symmetry analysis with the numerical methods. Two new reduced fractional differential equations are obtained by using the Lie symmetry theory. Applying only one Lie transformation, we get a new time-fractional partial differential equation and, applying a further Lie transformation, we get an ordinary differential equation. Numerical solutions of the reduced differential equations are computed separately by implicit numerical methods. A comparative study between numerical solutions is performed.

Invariance Analysis, Explicit Solution and Numerical Exact solution of Time Fractional Partial Differential Equation

In this research article, we have performed invariance analysis to time fraction partial differential equation (FPDE) by Lie symmetry reduction and converted the fractional order system into fractional ordinary differential equation (FODE) with the application of Erdyli-Kober (E-K) differ-integral operators in Riemann-Liouville (R-L) sense. Exact solutions are being established by power series technique and numerical solutions have been verified with application of the Fractional Reduced Differential Transforms and Homotop Analysis Methods.

Analytical Solution Of Time Fractional Nonlinear Schrodinger Equation By Homotopy Analysis Method

In this research paper, we have applied Homotopy Analysis Approach to the “time fractional nonlinear Schrodinger equation” to find its analytical periodic and solitary wave solution. Presence of convergence control parameter in this method guarantee the solution of time fractional differential equation in the form of rapidly convergent series. Obtained analytical solution has been compared and found in good agreement. This work demonstrates reliability and potential of HAM to study the time fractional partial differential equation

A time-fractional diffusion equation with space-time dependent hidden-memory variable order: analysis and approximation

A time-fractional partial differential equation with a hidden-memory space-time dependent variable order is studied. The well-posedness and high-order regularity estimates of the problem are proved via the resolvent estimates. Accordingly, an error estimate of the L-1 discretization without any artificial regularity assumption of the true solution is analyzed. Numerical experiments are performed to substantiate the theoretical results and to study the behavior of the variable-order fractional partial differential equations.

A robust numerical solution to a time-fractional Black\u2013Scholes equation

Dividend paying European stock options are modeled using a time-fractional Black–Scholes (tfBS) partial differential equation (PDE). The underlying fractional stochastic dynamics explored in this work are appropriate for capturing market fluctuations in which random fractional white noise has the potential to accurately estimate European put option premiums while providing a good numerical convergence. The aim of this paper is two fold: firstly, to construct a time-fractional (tfBS) PDE for pricing European options on continuous dividend paying stocks, and, secondly, to propose an implicit finite difference method for solving the constructed tfBS PDE. Through rigorous mathematical analysis it is established that the implicit finite difference scheme is unconditionally stable. To support these theoretical observations, two numerical examples are presented under the proposed fractional framework. Results indicate that the tfBS and its proposed numerical method are very effective mathematical tools for pricing European options.

Numerical solution for solving fractional parabolic partial differential equations

In this paper, A reliable numerical scheme is developed and reviewed in order to obtain approximate solution of time fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time fractional derivative is employed in the Caputo sense. TheL2convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of spectral method and justification the theoretical analysis, and to compare with other existing methods in the literatures

A new approximate analytical method and its convergence for nonlinear time-fractional partial differential equations

The main goal of this paper is to present a new approximate analytical method called modified generalized Taylor fractional series method (MGTFSM) for solving general nonlinear time- fractionalpartial differential equations. The fractional derivative is considered in the Caputo sense. We establish the convergence results of the proposed method. The basic idea of the MGTFSM is to construct the solution in the form of infinite series which converges rapidly to the exact solution of the given problem. The main advantage of the proposed method compare to current methods is that method solves the nonlinear problems without using linearization, discretization, perturbation or anyother restriction. The efficiency and accuracy of the MGTFSM is tested by means of different numerical examples. The results prove that the proposed method is very effective and simple for solving the nonlinear fractional problems.

A novel meshless collocation solver for solving multi-term variable-order time fractional PDEs

This paper presents a novel meshless collocation method to solve multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method, it employs the Fourier series expansion for spatial discretization, which transforms the original multi-term VOTFPDEs into a sequence of multi-term variable-order time fractional ordinary differential equations (VOTFODEs). Then, these VOTFODEs can be solved using the recent-developed backward substitution method. Several numerical examples verify the accuracy and efficiency of the proposed numerical approach in the solution of multi-term VOTFPDEs.

A modified numerical algorithm based on fractional Euler functions for solving time-fractional partial differential equations

ABSTRACT A novel and efficient method based on the fractional Euler function together with the collocation method is proposed to solve time-fractional partial differential equations. By applying the Riemann-Liouville fractional integral operator on this problem, we convert it to fractional partial integro-differential equations. Also, we present the method of calculating the operational matrix in a new way. These matrices with the help of the collocation method reduce the problem to a system of algebraic equations that can be easily solved by any usual numerical methods. Furthermore, we discuss error analysis for this method. Finally, the numerical technique is implemented for several examples to illustrate the superiority and efficiency of the proposed method in comparison with some other methods.

A superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity

AbstractIn this paper, a superlinear convergence scheme for the multi‐term and distribution‐order fractional wave equation with initial singularity is proposed. The initial singularity of the solution of the multi‐term time fractional partial differential equation often generate a singular source, it increases the difficulty to numerically solve the equation. Thus, after discretizing the spatial distribution‐order derivative by the midpoint quadrature, an integral transformation is applied to deal with the temporal direction for obtaining a temporal superlinear convergence scheme based on the uniform mesh. Then, the fully discrete scheme is constructed by using Crank–Nicolson technique and L1 approximation in time, and fractional centered difference approximation in space. The convergence and stability of the proposed scheme are rigorously analyzed. Finally, numerical experiments are presented to support the theoretical results of our scheme.

A semi-analytical method for solving a class of non-homogeneous time-fractional partial differential equations

In this paper, a new kind of hybrid method is established to get the analytical approximate solutions for a class of non-homogeneous time-fractional partial differential equations. This hybrid meth...

A semi-analytical method for solving a class of non-homogeneous time-fractional partial differential equations

A semi-analytical method for solving a class of non-homogeneous time-fractional partial differential equations

Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method

Abstract Multi-term time-fractional partial differential equations (PDEs) have become a hot topic in the field of mathematical physics and are used to improve the modeling accuracy in the description of anomalous diffusion processes compared to the single-term PDE results. This research includes the numerical solutions of two-term time-fractional PDE models using an efficient and accurate local meshless method. Due to the advantages of the meshless nature and ease of applicability in higher dimensions, the demand for meshless techniques is increasing. This approach approximates the solution on a uniform or scattered set of nodes, resulting in well-conditioned and sparse coefficient matrices. Numerical tests are performed to demonstrate the efficacy and accuracy of the proposed local meshless technique.