A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

A backward euler orthogonal spline collocation method for the time‐fractional Fokker–Planck equation

We formulate and analyze a novel numerical method for solving a time‐fractional Fokker–Planck equation which models an anomalous subdiffusion process. In this method, orthogonal spline collocation is used for the spatial discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1534–1550, 2015     

Numerical solution of the multiterm time‐fractional diffusion equation based on reproducing kernel theory

In this study, we present an efficient computational method for finding approximate solution of the multi term time‐fractional diffusion equation. The approximate solution is presented in the form of a finite series in a reproducing kernel Hilbert space. The convergence of proposed method is studied under some hypothesis which provides the theoretical basis of proposed method for solving the considered equation. Finally, some numerical experiments are considered to examine the efficiency of proposed method in the sense of accuracy and CPU time.     

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.     

High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem

We use the generalized L1 approximation for the Caputo fractional derivative, the second-order fractional quadrature rule approximation for the integral term, and a classical Crank-Nicolson alternating direction implicit (ADI) scheme for the time discretization of a new two-dimensional (2D) fractional integro-differential equation, in combination with a space discretization by an arbitrary-order orthogonal spline collocation (OSC) method. The stability of a Crank-Nicolson ADI OSC scheme is rigourously established, and error estimate is also derived. Finally, some numerical tests are given.     

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

In this paper, we study the numerical solution to time‐fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time‐fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Orthogonal spline collocation methods for the stream function‐vorticity formulation of the Navier–Stokes equations

An orthogonal spline collocation (OSC) spatial discretization is proposed for the solution of the fully coupled stream function‐vorticity formulation of the Navier–Stokes equations in two dimensions. For the time‐stepping, a three‐level leapfrog scheme is employed. This method is algebraically linear, and, at each time step, gives rise to a system of linear equations of the form arising in the OSC approximation of the biharmonic Dirichlet problem and can be solved by a fast direct method. Error estimates in the H^l–norm in space, l = 1,2, are derived for the semi‐discrete method and the fully‐discrete leapfrog scheme which is also shown to be second order accurate in time. Numerical results are presented which confirm the theoretical analysis and exhibit superconvergence phenomena, which provide superconvergent approximations to the components of the velocity. © John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A collocation method of lines for two‐sided space‐fractional advection‐diffusion equations with variable coefficients

We present the method of lines (MOL), which is based on the spectral collocation method, to solve space‐fractional advection‐diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left‐ and right‐sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method.     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014     

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.     

Finite difference scheme for time-space fractional diffusion equation with fractional boundary conditions

In this paper, the finite difference scheme is developed for the time-space fractional diffusion equation with Dirichlet and fractional boundary conditions. The time and space fractional derivatives are considered in the senses of Caputo and Riemann-Liouville, respectively. The stability and convergence of the proposed numerical scheme are strictly proved, and the convergence order is O(τ^2−α+h²). Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.     

High order convergent modified nodal bi-cubic spline collocation method for elliptic partial differential equation

A high order modified nodal bi-cubic spline collocation method is proposed for numerical solution of second-order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.     

B‐spline collocation algorithms for numerical solution of the RLW equation

Both sextic and septic B‐spline collocation algorithms are presented for the numerical solutions of the RLW equation. Numerical results resolve the fine structure of the single solitary wave propagation, two and three solitary waves interaction, and evolution of solitary waves. Comparison of the numerical results is done by the results of some earlier schemes mentioned in the article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 581–607, 2011     

Alternating direction implicit OSC scheme for the two-dimensional fractional evolution equation with a weakly singular kernel

In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.     

Multilevel numerical solutions of convection‐dominated diffusion problems by spline wavelets

In this article, we utilize spline wavelets to establish an adaptive multilevel numerical scheme for time‐dependent convection‐dominated diffusion problems within the frameworks of Galerkin formulation and Eulerian‐Lagrangian localized adjoint methods (ELLAM). In particular, we shall use linear Chui‐Quak semi‐orthogonal wavelets, which have explicit expressions and compact supports. Therefore, both the diffusion term and boundary conditions in the convection‐diffusion problems can be readily handled. Strategies for efficiently implementing the scheme are discussed and numerical results are interpreted from the viewpoint of nonlinear approximation. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

A fast second‐order difference scheme for the space–time fractional equation

In this paper, a fast second‐order accurate difference scheme is proposed for solving the space–time fractional equation. The temporal Caputo derivative is approximated by ?L2 ‐1_σ formula which employs the sum‐of‐exponential approximation to the kernel function appeared in Caputo derivative. The second‐order linear spline approximation is applied to the spatial Riemann–Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes.