Time Fractional Reaction Diffusion Equation Research Articles

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

Higher order numerical approximations for non-linear time-fractional reaction–diffusion equations exhibiting weak initial singularity

In the present study, we introduce a high-order non-polynomial spline method designed for non-linear time-fractional reaction–diffusion equations with an initial singularity. The method utilizes the L2-1σ scheme on a graded mesh to approximate the Caputo fractional derivative and employs a parametric quintic spline for discretizing the spatial variable. Our approach successfully tackles the impact of the singularity. The obtained non-linear system of equations is solved using an iterative algorithm. We provide the solvability of the novel non-polynomial scheme and prove its stability utilizing the discrete energy method. Moreover, the convergence of the proposed scheme has been established using the discrete energy method in the L2 norm. It is proven that the method is convergent of order min{θα,2} in the temporal direction and 4.5 in the spatial direction, where α∈(0,1) denotes the order of the fractional derivative and the parameter θ is utilized in the construction of the graded mesh. Finally, we conduct numerical experiments to validate our theoretical findings and to illustrate how the mesh grading influences the convergence order when dealing with a non-smooth solution to the problem.

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

Nonlinear two-component system of time-fractional PDEs in [formula omitted]-dimensions: Invariant subspace method combined with variable transformation

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite-dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which again reduces to a system of time-fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two-component system of time-fractional reaction–diffusion equations (TFRDEs) in (2+1)-dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

Determination of Timewise-Source Coefficient in Time-Fractional Reaction-Diffusion Equation from First Order Heat Moment

This article aims to determine the time-dependent heat coefficient together with the temperature solution for a type of semi-linear time-fractional inverse source problem by applying a method based on the finite difference scheme and Tikhonov regularization. An unconditionally stable implicit finite difference scheme is used as a direct (forward) solver. While by the MATLAB routine lsqnonlin from the optimization toolbox, the inverse problem is reformulated as nonlinear least square minimization and solved efficiently. Since the problem is generally incorrect or ill-posed that means any error inclusion in the input data will produce a large error in the output data. Therefore, the Tikhonov regularization technique is applied to obtain stable and accurate results. Finally, to demonstrate the accuracy and effectiveness of our scheme, two benchmark test problems have been considered, and its good working with different noise levels.

A Mixed Finite Element Method for the Multi-Term Time-Fractional Reaction–Diffusion Equations

In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete L∞(L2(Ω)) norm and for the auxiliary variable λ in the discrete L∞((L2(Ω))2) and L∞(H(div,Ω)) norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results.

L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations

L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations

On the soliton solutions to the density-dependent space time fractional reaction–diffusion equation with conformable and M-truncated derivatives

On the soliton solutions to the density-dependent space time fractional reaction–diffusion equation with conformable and M-truncated derivatives

Comparative analysis of three memory selection methods for time integration of Fractional Reaction-Diffusion Equations

By discretising in space, a non-linear time fractional reaction-diffusion equations (TFRDEs) can be converted into a system of time-fractional differential equations (TFDEs). The full memory method (FMM) and short memory method (SMM) are well-established memory selection methods used in the time integration of TFDEs. The main drawbacks of FMM and SMM are higher computational cost and uncontrollable error respectively. The only way to increase the accuracy of SMM is by increasing short memory length which causes an increase in computational cost. Especially when we apply these two methods to integrate TFRDEs, we have to solve a large system of TFDEs. Therefore, the drawbacks of these two methods affect seriously, when these are applied to solve TFRDEs. This paper aims to investigate the accuracy and efficiency of the memory selection method, Exponentially Decreasing Random Memory Method (EDRMM), and compare it with FMM and SMM when these methods apply to integrate TFRDEs. Based on these three memory selection methods, three semi-implicit numerical schemes namely semiimplicit scheme with full memory method (SI-FMM), semi-implicit scheme with short memory method (SI-SMM), and semi-implicit scheme with exponentially decreasing random memory method (SI-EDRMM)) are proposed and the accuracy and CPU time (computational time (CT)) of these three numerical schemes are compared. To do this comparison, these three numerical schemes are applied to four TFRDEs whose exact solutions are known. Numerical experiments confirm that the accuracy and efficiency of the SI-EDRMM are better than that of SI-SMM and the efficiency of SI-EDRMM is higher than that of SI-FMM. Therefore, EDRMM is better than SMM and FMM for the integration of TFRDEs.

Interpolated Coefficient Mixed Finite Elements for Semilinear Time Fractional Diffusion Equations

In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic L1 scheme based on graded meshes and new mixed finite element based on triangulation is used for the temporal and spatial discretization, respectively. The interpolation coefficient technique is used to deal with the semilinear term, and the discrete nonlinear system is solved by a Newton-like iterative method. Stability and convergence results for both the original variable and its flux are derived. Numerical experiments confirm our theoretical analysis.

Local Error Estimate of an L1-Finite Difference Scheme for the Multiterm Two-Dimensional Time-Fractional Reaction–Diffusion Equation with Robin Boundary Conditions

In this paper, the numerical method for a multiterm time-fractional reaction–diffusion equation with classical Robin boundary conditions is considered. The full discrete scheme is constructed with the L1-finite difference method, which entails using the L1 scheme on graded meshes for the temporal discretisation of each Caputo fractional derivative and using the finite difference method on uniform meshes for spatial discretisation. By dealing with the discretisation of Robin boundary conditions carefully, sharp error analysis at each time level is proven. Additionally, numerical results that can confirm the sharpness of the error estimates are presented.

A fully discrete scheme based on cubic splines and its analysis for time-fractional reaction–diffusion equations exhibiting weak initial singularity

The aim of this paper is to design and analyze a robust fully discrete scheme based on cubic splines for numerically solving a time-fractional reaction–diffusion equation (TFRDE) with smooth and non-smooth solutions. The solution of this problem exhibits a weak singularity near time t=0. The method combines the L1 scheme on a uniform or graded mesh to discretize in time and a cubic spline difference scheme on a uniform mesh to discretize in space. Further, the stability and convergence for both the smooth and non-smooth solutions are analyzed separately. Moreover, due to the use of a cubic spline difference scheme to discretize the space variable, the analysis of the fully discrete scheme requires some innovative ideas. In the end, two test problems are considered to demonstrate the validity and authenticity of the proposed method. The first example deals with the smooth solution, whereas in the second example, we consider a non-smooth solution.

A fourth order accurate numerical method for non-linear time fractional reaction–diffusion equation on a bounded domain

In the present work, a high-order numerical scheme is proposed and analyzed to solve the non-linear time fractional reaction–diffusion equation (RDE) of order α∈(0,1). The numerical scheme consists of a time-stepping cubic approximation for the time fractional derivative and a compact finite difference scheme to approximate the spatial derivative. After applying these approximations to time fractional RDE, we get a non-linear system of equations. An iterative algorithm is formulated to solve the obtained nonlinear discrete system. We analyze the unique solvability of the proposed compact finite difference scheme and discuss the stability using von Neumann analysis. Further, we prove that the scheme is convergent in the Euclidean norm with the convergence order 4−α in the temporal direction and 4 in the spatial direction using matrix analysis. Finally, the numerical experimentation is performed to demonstrate the authenticity of the proposed numerical scheme.

Green’s Functions on Various Time Scales for the Time-Fractional Reaction-Diffusion Equation

The time-fractional diffusion equation coupled with a first-order irreversible reaction is investigated by employing integral transforms. We derive Green’s functions for short and long times via approximations of the Mittag-Leffler function. The time value for which the crossover between short- and long-time asymptotic holds is presented in explicit form. Based on the developed Green’s functions, the exact analytic asymptotic solutions of the time-fractional reaction-diffusion equation are obtained. The applicability of the obtained solutions is demonstrated via quantification of the reaction-diffusion kinetics during heterogeneous catalytic chitin conversion to chitosan.

Laplace-Residual Power Series Method for Solving Time-Fractional Reaction–Diffusion Model

Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the nonlinear time-fractional reaction–diffusion model. The new approach is called the Laplace-residual power series method (L-RPSM), which imitates the residual power series method in determining the coefficients of the series solution. The proposed method is also adapted to find an approximate series solution that converges to the exact solution of the nonlinear time-fractional reaction–diffusion equations. In addition, the method has been applied to many examples, and the findings are found to be impressive. Further, the results indicate that the L-RPSM is effective, fast, and easy to reach the exact solution of the equations. Furthermore, several actual and approximate solutions are graphically represented to demonstrate the efficiency and accuracy of the proposed method.

Glioblastoma multiforme growth prediction using a Proliferation-Invasion model based on nonlinear time-fractional 2D diffusion equation

The glioblastoma multiforme (GBM) is the fast-growing and aggressive type of tumor of the brain or spinal cord that is associated with high morbidity and mortality. Therefore, accurate knowledge of the tumor growth process in different time intervals seems necessary to adopt optimal treatment methods. In this study, with a combination of available early stages imaging data and using image processing methods, a time fractional reaction–diffusion equation based on the moving least squares method was implemented for predicting the growth of the GBM tumor model implemented in the mouse brain. The obtained results demonstrate that the Proliferation-Invasion model implemented by a time fractional form shows more agreement with the experimental results. As a result, using a fractional derivative of order α=0.9 results in the lowest prediction error of the tumor surface (less than 1%). In addition, to reducing the cost and side effects of theranostic methods, the presented model will have the possibility to consider the effects of therapeutic factors, such as hyperthermia, radiography, and surgery, on tumor growth. The ability to use the patient’s characteristics in diagnostic and treatment considerations and the development of personalized medicine can also be considered one of the capabilities of the presented model.

A new fast predictor-corrector method for nonlinear time-fractional reaction-diffusion equation with nonhomogeneous terms

Time-fractional reaction-diffusion (TFRD) equation is an important fractional parabolic equation, and the research of its numerical solution has scientific significance and engineering application value. In this paper, a new fast predictor-corrector (FP-C) scheme is constructed based on fast L1 approximation of Caputo fractional derivative for solving nonlinear TFRD equation with nonhomogeneous terms. The linearized implicit difference scheme is used for the predictor step and Crank-Nicolson(C-N) scheme is used for the corrector step. Theoretical analysis proves that FP-C scheme is convergent and stable unconditionally for nonlinear TFRD equation. Numerical analysis and experiments show that the computational accuracy of FP-C scheme is $ O(\tau^{2-\alpha}+h^2) $ under the strong regularity condition and $ O(\tau^{\alpha}+h^2) $ under the weak regularity condition. Compared with the classical predictor-corrector (P-C) scheme based on standard L1 approximation, the FP-C scheme improves the computational efficiency without losing the computational accuracy. It is shown that the new FP-C scheme is an efficient method to solve nonlinear TFRD equation with nonhomogeneous terms.

A capable numerical meshless scheme for solving distributed order time-fractional reaction–diffusion equation

Distributed order fractional differential equations are efficient in describing physical phenomena because of the differential order distribution. In this paper, the distributed order time-fractional reaction–diffusion equation is considered in the sense of Caputo fractional derivative. A hybrid method is developed based on the Moving Kriging (MK) interpolation and finite difference method for solving this distributed order equation. First, the distributed integral is discretized by the M-point Gauss Legendre quadrature rule. Then, the L2−1σ method is applied to approximate the solution of the fractional derivative discretization. Also, the unconditionally stability and rate of convergence O(τ2) of the time-discrete technique are illustrated. Furthermore, the MK interpolation is resulted in the space variables discretization. Finally, some examples are presented to indicate the efficiency of this method and endorsement the theoretical results.