A new extended cubic B-spline approximation for the numerical solution of the time-fractional fisher equation is formed and examined. The given non-linear partial differential equation through substitution is converted into a linear partial differential equation through substitution, using Taylor's series expansion. The time-fractional derivative is approximated in Caputo's sense while the space dimension is calculated using a new extended cubic B-spline. The proposed numerical technique is shown to be unconditionally stable and convergent. The errors are used for measuring the accuracy of the proposed technique. The graphical and numerical results are presented to illustrate the performance of the technique.

The time–fractional reaction–diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo–Fabrizio operator is applied for the discretization of time levels while the extended cubic B-spline (ECBS) function is applied for the space direction. The ECBS function preserves geometrical invariability, convex hull and symmetry property. Unconditional stability and convergence analysis are also proved. The projected numerical method is tested on two numerical examples. The theoretical and numerical results demonstrate that the order of convergence of 2 in time and space directions.

In this paper, new group iterative schemes are developed for the numerical solution of two-dimensional anomalous fractional sub-diffusion equation subject to specific initial and Dirichlet boundary conditions. The new group relaxation iterative schemes are derived from the combination of standard and rotated (skewed) five-point modified implicit finite difference approximations. The results derived from the conducted numerical experiments show that fractional explicit de-coupled group (FEDG) iterative method has a significantly less computational cost in terms of CPU-timings as compared to the other iterative schemes, without threatening compromising accuracies.

This paper presents the development of a new iterative method for solving the two-dimensional hyperbolic telegraph fractional differential equation (2D-HTFDE) which is central to the mathematical modeling of transmission line satisfying certain relationship between voltage and current waves in specific distance and time. This equation can be obtained from the classical two-dimensional hyperbolic telegraph partial differential equation by replacing the first and second order time derivatives by the Caputo time fractional derivatives of order 2α and α respectively, with 1/2<α<1$1/2< \alpha < 1 $. The iterative scheme, called the fractional skewed grid Crank–Nicolson FSkG(C-N), is derived from finite difference approximations discretized on a skewed grid rotated clockwise 450 from the standard grid. The skewed finite difference scheme combined with Crank–Nicolson discretization formula will be shown to be unconditionally stable and convergent by the Fourier analysis. The developed FSkG(C-N) scheme will be compared with the fractional Crank–Nicolson scheme on the standard grid to confirm the effectiveness of the proposed scheme in terms of computational complexities and computing efforts. It will be shown that the new proposed scheme demonstrates more superior capabilities in terms of the number of iterations and CPU timings compared to its counterpart on the standard grid but with the same order of accuracy.

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions.