Time Fractional Cable Equation Research Articles

Utilizing differential quadrature-based RBF partition of unity collocation method to simulate distributed-order time fractional Cable equation

Utilizing differential quadrature-based RBF partition of unity collocation method to simulate distributed-order time fractional Cable equation

Stability and Convergence of L1-Galerkin Spectral Methods for the Nonlinear Time Fractional Cable Equation

Stability and Convergence of L1-Galerkin Spectral Methods for the Nonlinear Time Fractional Cable Equation

The direct meshless local Petrov–Galerkin technique with its error estimate for distributed-order time fractional Cable equation

Distributed-order fractional calculus is a quickly growing concept of the more general area of fractional calculus that has significant and extensive usage for designing complex systems. This work is used the direct meshless local Petrov–Galerkin (DMLPG) technique for the numerical solution of the distributed-order time fractional Cable equation. DMLPG implements a generalized moving least square (GMLS) process to discretize the equation in space variables. By using this scheme, the test function is approximated via the values at nodes, directly. Thus, this algorithm passes integration with the MLS shape functions substituting with a more inexpensive integration than polynomials. Here, the distributed integral is discretized by the M-point Gauss–Legendre quadrature rule. Then, the finite difference scheme is applied to approximate the fractional derivative discretization. Also, the unconditionally stability and rate of convergence O(τ2−max{α,β}) of the time-discrete technique are demonstrated. Moreover, the current method converts the problem into a system of linear algebraic equations. To demonstrate the capability and flexibility of our scheme, some examples with different geometric domains are supposed in two-dimensional cases.

A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation

In this article, we developed a new higher-order implicit finite difference iterative scheme (FDIS) for the solution of the two dimension (2-D) time fractional Cable equation (FCE). In the new proposed FDIS, the time fractional and space derivatives are discretized using the Caputo fractional derivative and fourth-order implicit scheme, respectively. Moreover, the proposed scheme theoretical analysis (convergence and stability) is also discussed using the Fourier analysis method. Finally, some numerical test problems are presented to show the effectiveness of the proposed method.

A meshless method for solving two-dimensional distributed-order time-fractional cable equation

In this paper, a meshless method is proposed for solving two-dimensional distributed-order time-fractional cable equation. This method uses the basis functions that constructed by Gauss kernel as shape functions. Then we introduce the minimum residual approximate solution to get approximate solution and provide error estimation under the condition of well-posed equations. Finally, we provide two numerical examples to verify the accuracy of our method.

Error Bounds of a Finite Difference/Spectral Method for the Generalized Time Fractional Cable Equation

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have min{2−α,2−β}-order convergence in time and spectral accuracy in space for smooth solutions, where α,β are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow.

The design of new high-order group iterative method in the solution of two-dimensional fractional cable equation

In this article, a new high-order explicit group iterative scheme is developed for the solution of the two-dimensional time fractional cable equation. The proposed scheme is derived from the Crank-Nicolson (C-N) high-order compact finite difference method, where the Caputo discretization and C-N high-order approximations are used for the time fractional and space derivative respectively. The stability and convergence of the proposed scheme are also established. Finally, two numerical examples are presented to show the accuracy and efficiency of the scheme.

Finite difference/spectral methods for the two-dimensional distributed-order time-fractional cable equation

The cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics. In recent years, considerable attention has been devoted to distributed-order differential equations because they appear to be more effective for modeling complex processes. In this work, a finite difference/Legendre spectral method is presented for the numerical simulation of the two-dimensional (2D) distributed-order time-fractional cable equation, where the finite difference method is employed in the temporal discretization and Legendre spectral method is adopted in the spatial discretization. The midpoint quadrature rule is used to approximate the distributed-order, such that the considered equation could be transformed into a multi-term fractional equation. The stability and convergence analysis of the proposed scheme is established, which illustrates that the numerical solution converges to the exact solution with order O(τ2+σ2+N−s), where τ, σ, N are the time step size, the step length in the approximation of the distributed-order and the polynomial degree, respectively. Furthermore, to demonstrate the versatility and applicability of our method, we provide numerical results that show good agreement with the theoretical analysis.

Efficient hybrid group iterative methods in the solution of two-dimensional time fractional cable equation

In this paper, the development of new hybrid group iterative methods for the numerical solution of a two-dimensional time-fractional cable equation is presented. We use Laplace transform method to approximate the time fractional derivative which reduces the problem into an approximating partial differential equation. The obtained partial differential equation is solved by four-point group iterative methods derived from two implicit finite difference schemes. Matrix norm analysis together with mathematical induction are utilized to investigate the stability and convergence properties. A comparative study with the recently developed hybrid standard point (HSP) iterative method accompanied by their computational cost analysis are also given. Numerical experiments are conducted to demonstrate the superiority of the proposed hybrid group iterative methods over the HSP iterative method in terms of the number of iterations, computational cost as well as the CPU times.

A novel finite element method for the distributed-order time fractional Cable equation in two dimensions

In this paper, the unstructured mesh Galerkin finite element method with a weighted and shifted Grünwald difference approximation and Composite Trapezoid formula is presented to solve the nonhomogeneous two-dimensional distributed order time fractional Cable equation on irregular convex domains. The Crank–Nicolson type discretization of the finite element scheme is implemented to obtain the numerical solution. The stability and convergence of the numerical scheme are discussed and derived. Finally, some numerical examples on irregular convex domains are given to confirm our theoretical results.

Numerical solution of two‐dimensional time fractional cable equation with Mittag‐Leffler kernel

The main motive of this article is to study the recently developed Atangana‐Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag‐Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two‐dimensional cable equation in which time‐fractional derivative is of Mittag‐Leffler kernel type. First, we derive an approximation formula of the fractional‐order ABC derivative of a function tk using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two‐dimensional model of the time‐fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two‐dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag‐Leffler fractional derivative.

Numerical Solution of Time Fractional Cable Equation via the Sinc-Bernoulli Collocation Method

An important equation usually used in modeling neuronal dynamics is cable equation. In this work, a numerical method for the fractional cable equation which involves two Riemann-Liouville fractional derivatives is proposed. Our computational technique is based on collocation idea where a combination of Bernoulli polynomials and Sinc functions are used to approximate the solution to this problem. The constructed approximation by our method convert the fractional cable equation into a set of algebraic equations. Also, we provide two numerical examples to confirm the accuracy and effectiveness of the present method.

A novel numerical method for solving the 2-D time fractional cable equation

In this paper, an efficient numerical scheme is constructed to approximate the solutions of the two-dimensional linear and nonlinear fractional cable equation. The fractional derivative is defined in the Atangana-Baleanu-Caputo sense. The nonstandard implicit compact finite difference method is presented to solve numerically the proposed equation. Special attention is given to studying the stability analysis of the numerical technique. Moreover, the truncation error is analyzed. Finally, two numerical test examples and comparative studies are presented. The numerical experiments demonstrate the high accuracy of the new method.

Galerkin spectral method for nonlinear time fractional Cable equation with smooth and nonsmooth solutions

In this work, we study the numerical solutions of the time fractional Cable equations with nonlinear term, where the fractional derivatives are described in Riemann–Liouville sense. An explicit scheme is constructed based upon finite difference method in time and Legendre spectral method in space. Stability and convergence of scheme are proved rigorously. Moreover, an improved algorithm for the problem with nonsmooth solutions is proposed by adding correction terms to the approximations of first-order derivative, fractional derivatives and nonlinear term. Numerical examples are given to support theoretical analysis.

The lumped mass FEM for a time-fractional cable equation

We consider the numerical approximation of a time-fractional cable equation involving two Riemann–Liouville fractional derivatives. We investigate a semidiscrete scheme based on the lumped mass Galerkin finite element method (FEM), using piecewise linear functions. We establish optimal error estimates for smooth and middly smooth initial data, i.e., v∈Hq(Ω)∩H01(Ω), q=1,2. For nonsmooth initial data, i.e., v∈L2(Ω), the optimal L2(Ω)-norm error estimate requires an additional assumption on mesh, which is known to be satisfied for symmetric meshes. A quasi-optimal L∞(Ω)-norm error estimate is also obtained. Further, we analyze two fully discrete schemes using convolution quadrature in time based on the backward Euler and the second-order backward difference methods, and derive error estimates for smooth and nonsmooth data. Finally, we present several numerical examples to confirm our theoretical results.

A time–space spectral tau method for the time fractional cable equation and its inverse problem

This paper is devoted to the numerical solution of the time fractional cable equation and its inverse problem. The time–space spectral Legendre tau method based on the shifted Legendre polynomial and its operational matrices is used to solve the direct problem. Furthermore, we prove that the approximated solution of this method converges to the exact solution. In addition, the inverse problem is formulated by using the Tikhonov regularization, the stability and convergence for the inverse problem are provided, then we analyse the sensitivity coefficients and apply the nonlinear conjugate gradient method to solve the regularized problem. Finally, some numerical results are carried out to support the theoretical claims.

Numerical simulation of time fractional Cable equations and convergence analysis

In this article, the numerical solution of time fractional Cable equation is considered. We convert the time fractional Cable equations into equivalent integral equations with singular kernel, then propose a spectral collection method in both time and space discretizations with a spectral expansion of Lagrange interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results.

Time fractional cable equation and applications in neurophysiology

We propose an extension of the cable equation by introducing a Caputo time fractional derivative. The fundamental solutions of the most common boundary problems are derived analytically via Laplace Transform, and result be written in terms of known special functions. This generalization could be useful to describe anomalous diffusion phenomena with leakage as signal conduction in spiny dendrites. The presented solutions are computed in Matlab and plotted.

The use of local radial point interpolation method for solving two-dimensional linear fractional cable equation

An efficient numerical technique is formulated to solve two-dimensional time fractional cable equation. The fractional cable equation is an important mathematical model for describing anomalous diffusion processes in biological systems. The proposed computational technique is based on the combination of time stepping method and meshless weak formulation. At the first step, some implicit difference schemes are used to discrete the appearing integer and fractional time derivatives. Then, local radial point interpolation method (LRPI) is extended and used to solve the semi-discretized problem. The main aim of the paper is to show that the LRPI method is a powerful alternative computational technique to solve complicated fractional problems with high accuracy and low complexity. The performance and accuracy of the method are studied and verified through numerical experiments. Moreover, the convergence rate of the temporal discretization scheme is investigated numerically.

A two-grid finite element approximation for a nonlinear time-fractional Cable equation

In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Gr\"unwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved, Second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time is proved in detail, which is higher than the L1-approximation result with $O(\tau^{2-\alpha}+\tau^{2-\beta})$. Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU-time is compared to verify our theoretical results.