Space-time Fractional Partial Differential Equations Research Articles

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

PurposeFractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.Design/methodology/approachThis paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.FindingsThis paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.Originality/valueThis paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

Solutions of Time-Space Fractional Partial Differential Equations Using Picard's Iterative Method

Abstract In this paper, we present Picard's iterative method (PIM) for solving time–space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The proposed method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.

The modified generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Schrödinger type

This paper presents a modified version of the generalized Kudryashov method aimed at obtaining exact solutions for fractional partial differential equations of Schrödinger type. Firstly, the fractional partial differential equations are transformed into a set of ordinary differential equations using the fractional complex transform. Subsequently, a modified version of the generalized Kudryashov method is employed to solve these equations and the obtained results are utilized to derive the traveling wave solutions for fractional partial differential equations of the Schrödinger type. Finally, the graphs resulting from the traveling wave solution are examined, encompassing the analysis of kink waves and singular kink waves.

Exact Solutions of Systems of Nonlinear Time-Space Fractional Partial Differential Equations Using an Iterative Method

Abstract Fractional partial differential equations are useful tools to describe transportation, anomalous, and non-Brownian diffusion. In the present paper, we propose the Daftardar-Gejji and Jafari method along with its error analysis for solving systems of nonlinear time–space fractional partial differential equations (PDEs). Moreover, we solve a variety of nontrivial time–space fractional systems of PDEs. The obtained solutions either occur in exact form or in the form of a series, which converges to a closed form. The proposed method is free from linearization and discretization and does not include any tedious calculations. Moreover, it is easily employable using the computer algebra system such as Mathematica, Maple, etc.

Numerical and Analytical Solutions of Space-Time Fractional Partial Differential Equations by Using a New Double Integral Transform Method

This work discusses the beginning of fractional calculus and how the Sumudu and Elzaki transforms are applied to fractional derivatives. This approach combines a double Sumudu-Elzaki transform strategy to discover analytic solutions to space-time fractional partial differential equations in Mittag-Leffler functions subject to initial and boundary conditions. Where this method gets closer and closer to the correct answer, and the technique's efficacy is demonstrated using numerical examples performed with Matlab R2015a.

LIE SYMMETRY, EXACT SOLUTIONS AND CONSERVATION LAWS OF SOME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper, Lie symmetry analysis method is applied to spacetime fractional reaction-diffusion equations and diffusion-convection Boussinesq equations. The Lie symmetries for the governing equations are obtained and used to get group generators for reducing the space-time fractional partial differential equations(FPDEs) with Riemann-Liouville fractional derivative to space-time fractional ordinary differential equations(FODEs) with ErdélyiKober fractional derivative. Then the Laplace transformation and the power series methods are applied to derive explicit solutions for the reduced equations. Moreover, the conservation theorems and the generalization of the Noether operators are developed to acquire the conservation laws for the equations. Some figures for the obtained explicit solutions are also presented.

Interpolating Scaling Functions Tau Method for Solving Space–Time Fractional Partial Differential Equations

This paper is devoted to an innovative and efficient technique for solving space–time fractional differential equations (STFPDEs). To this end, we apply the Tau method such that the bases used are interpolating scaling functions (ISFs). The operational metrics for the derivative operator and fractional integration operator are used to introduce the operational matrix for the Caputo fractional derivative. Due to some characteristics of ISFs, such as interpolation, computation costs can be significantly reduced. We investigate the convergence of the technique, and some numerical implementations show that the method is effective for solving such equations.

MODIFIED FINITE ELEMENT NUMERICAL METHOD FOR SOLVING CONFORMABLE SPACE-TIME FRACTIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

This paper shows how to approximate the solution to a nonlinear conformable space-time fractional partial differential equations. The proposed method is based on the Cubic B-spline polynomials and Galerkin method. Two test problems show that the approach we use to approximate the proposed equation is accurate and efficient. We apply the Von Neumann approach to show that stability requires some conditions.

Diverse Precise Traveling Wave Solutions Possessing Beta Derivative of the Fractional Differential Equations Arising in Mathematical Physics

In this paper, we obtain the novel exact traveling wave solutions in the form of trigonometric, hyperbolic and exponential functions for the nonlinear time fractional generalized reaction Duffing model and density dependent fractional diffusion-reaction equation in the sense of beta-derivative by using three fertile methods, namely, Generalized tanh (GT) method, Generalized Bernoulli (GB) sub-ODE method, and Riccati-Bernoulli (RB) sub-ODE method. The derived solutions to the aforementioned equations are validated through symbolic soft computations. To promote the vital propagated features; some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the specific values to the parameters under the confine conditions. The accomplished solutions show that the presented methods are not only powerful mathematical tools for generating more solutions of nonlinear time fractional partial differential equations but also can be applied to nonlinear space-time fractional partial differential equations.

Consistent travelling waves solutions to the non-linear time fractional Klein–Gordon and Sine-Gordon equations through extended tanh-function approach

In this study, the extended tanh-function method has been used to find further general travelling wave solutions for space-time fractional nonlinear partial differential equations, namely, the time fractional nonlinear Sine-Gordon equation and Klein–Gordon equation. The mentioned equations are useful for explaining a series of experimental seismic data, modelling of strain waves, and structures correlated to fault dynamics and the subduction slab, as well as slow earthquakes, episodic tremor, slow slip events, and tremor migration patterns. This method is used to attain exact solutions to a variety of nonlinear partial differential equations that were spatially temporal fractional. To get analytical solutions of the travelling wave type of certain nonlinear evolution equations, a power series in tanh was first utilized as an ansatz. These nonlinear fractional partial differential equations (NLFPDEs) are solved on a variety of non-rectangular domains. By using the fractional complex transform and the properties of the confirmable derivative, the two equations are reduced to ordinary differential equations. The solutions are sketched in 3D, and contour patterns, including king type, single soliton, double solitons, multiple solitons, bell shape, and other sorts of solutions. The achieved solutions are very much effective for explaining the above-stated phenomena.

Exact solutions of (1+2)-dimensional non-linear time-space fractional PDEs

PurposeIn this paper, the author presents a hybrid method along with its error analysis to solve (1+2)-dimensional non-linear time-space fractional partial differential equations (FPDEs).Design/methodology/approachThe proposed method is a combination of Sumudu transform and a semi-analytc technique Daftardar-Gejji and Jafari method (DGJM).FindingsThe author solves various non-trivial examples using the proposed method. Moreover, the author obtained the solutions either in exact form or in a series that converges to a closed-form solution. The proposed method is a very good tool to solve this type of equations.Originality/valueThe present work is original. To the best of the author's knowledge, this work is not done by anyone in the literature.

Modification of numerical algorithm for space-time fractional partial differential equations including two types of fractional derivatives

This paper derives the numerical algorithm for a novel class of space-time fractional partial differential equations (STFPDEs), which include Atangana-Baleanu-Caputo (ABC) and Caputo fractional derivatives. To deal with the problems, we present the novel approximation tool based on shifted Gegenbauer polynomials (SGPs). The accuracy of the numerical method is sensitive to the accuracy of the operational matrices. Therefore, we improve the algorithm of calculating the operational matrix of integration and derivatives. By using the modified operational matrices and discretization method, the governing STFPDEs including different types of fractional derivatives are transformed into a system of algebraic equations. Furthermore, we investigate the error analysis of the method. Finally, several numerical examples are provided to verify the efficiency and accuracy of the method. From a computational point of view, the approximate solutions obtained from the method are in excellent agreement with the exact solution.

Analytical Solution of Nonlinear Nonhomogeneous Space and Time Fractional Physical Models by Improved Adomian Decomposition Method

The paper aims to obtain exact analytical solution of nonlinear nonhomogeneous space-time fractional order partial differential equations in Gas dynamics model, Advection model, Wave model and Klein-Gordon model by improved Adomian decomposition method coupled with fractional Taylor expansion series.The solution of these equations are in series form may have rapid convergence to a closed-form solution. The effectiveness and sharpness of this method is shown by obtaining the exact solution of these equations with suitable initial conditions(ICs). With the help of this method, it is possible to investigate nature of solutions when we vary order of the fractional derivative. Behaviour of the solution of these equations are represented by graphs using MATHEMATICA software.

<i>ψ</i>-SHIFTED OPERATIONAL MATRIX SCHEME FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

<abstract> In this paper, we present a numerical method to solve space-time fractional partial differential equations. We introduce $ \psi$-shifted Chebyshev polynomials to construct operational matrices of fractional differentiation in the Caputo sense. These operational matrices are then used to find the solution of fractional partial differential equations. The efficiency and applicability of introduced numerical scheme is tested by comparing the proposed numerical approximations with the results obtained from existing numerical methods. </abstract>

Some properties of space-time fractional stochastic partial differential equations with Lévy noise

We consider a class of space-time fractional stochastic partial differential equation on a bounded domain with Lévy noise. We prove that the second moment of the solution u(t,x) can not decay exponentially and for β∈(0,12], supx∈D𝔼|u(t,x)|2 grows exponentially fast for large t. When β∈(12,1), there is some phase transition of the second moment growth, depending on the noise level λ.

NUMERICAL SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS VIA LAPLACE TRANSFORM

In this study, solutions of time-space fractional partial differential equations(FPDEs) are obtained by utilizing the Shehu transform iterative method. The utilityof the technique is shown by getting numerical solutions to a large number of FPDEs.

A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

This paper is devoted to introducing a novel methodology to prove the convergence and stability of a Crank–Nicolson difference approximation for a class of multi-term time-fractional diffusion equations with nonlinear delay and space fractional derivatives in case of sufficient smooth solutions. The temporal fractional derivatives are approximated by a specific form of L1 scheme at tk+1/2. A fourth-order difference approximation for the spatial fractional derivatives is employed by using the weighted average of the shifted Grünwald formulae. This methodology is based on a class of discrete fractional Grönwall inequalities convenient with the quadrature formula used to approximate the Caputo derivative at tk+1/2. In the present work, the method of energy inequalities is utilized to show that the used difference scheme is stable and converges to the exact solution with order O(τ2−αJ+h4), in the case that 0<αJ<1 satisfies 3αJ≥32, which means that 0.369≤αJ<1, such that αJ is the maximum α-th order in the multi-order fractional operators.

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

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

Exact solutions of the fractional Sharma-Tasso-Olver equation and the fractional Bogoyavlenskii’s breaking soliton equations

In this work, we adopt the generalized Kudryashov method to produce exact wave solutions for some space-time fractional partial differential equations. The Jumaries modified Riemann-Liouville is considered. The proposed method is implemented to the fractional Sharma-Tasso-Olver equation and the fractional Bogoyavlenskii’s breaking soliton equations. Various solutions containing hyperbolic tangent function, trigonometric tangent function and exponential functions are obtained by this method. Simulations of some obtained solutions are given at the end of the paper.

Maximum Principle for Space and Time-Space Fractional Partial Differential Equations

In this paper we obtain new estimates of the sequential Caputo fractional derivatives of a function at its extremum points. We derive comparison principles for the linear fractional differential equations, and apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion possesses at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Luchko in 2011. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved.