High‐order compact finite difference and laplace transform method for the solution of time‐fractional heat equations with dirchlet and neumann boundary conditions

Abstract

The work presents a novel coupling of the Laplace Transform and the compact fourth-order finite-difference discretization scheme for the efficient and accurate solution of linear time-fractional nonhomogeneous diffusion equations subject to both Dirichlet and Neumann boundary conditions. A translational transformation of the dependent variable ensures the Caputo derivative is aligned with the Riemann-Louiville fractional derivative. The resulting scheme is computationally efficient and shown to be uniquely solvable in all cases, accurate and convergent to O( x 4 ) in the spatial domain. The convergence rates in the temporal domain are contour dependent but exhibit geometric convergence.

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

“…Remark Only simple homogeneous Dirichlet boundary conditions have been treated. However, many problems require more than this, for instance, nonstandard boundary conditions . In this case, to impose the boundary conditions, it is required to add additional equations rather than removing them , Chapter 13].…”

Numerical Solution of Fractional Order Anomalous Subdiffusion Problems Using Radial Kernels and Transform

“…Recently, Laplace transform is combined with RBF method in [32,33]. In [34][35][36][37], the authors use Laplace transform as tool in spectral method and other mesh-based methods such as finite element methods and finite difference method. To avoid the issues of computational efficiency and instability of the system matrix, we introduce a new technique Laplace transform-based local RBF method in solving the time fractional modified anomalous subdiffusion equations in irregular domain.…”

The Rational Spectral Method Combined with the Laplace Transform for Solving the Robin Time-Fractional Equation

“…To solve such problems, we need to introduce special functions to express the exact solutions of fractional differential equations, which can be very difficult. Therefore, attempts have been made to propose numerical methods that approximate the solutions of such equations [4][5][6][7][8][9].…”