Parameter uniform numerical methods for singularly perturbed delay parabolic differential equations with non-local boundary condition

Abstract

The motive of this paper is, to develop accurate and parameter uniform numerical method for solving singularly perturbed delay parabolic differential equation with non-local boundary condition exhibiting parabolic boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layer. Fitted operator finite difference method on uniform mesh that uses the procedures of method of line for spatial discretization and backward Euler method for the resulting system of initial value problems in temporal direction is considered. To treat the non-local boundary condition, Simpsons rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter. The method is shown to be accurate of O(h2 + △t) . Finally, conclusion of the work is included at the end.

Adaptive Technique for Solving 1-D Interface Problems of Fractional Order

“…The researchers applied the RKHS to develop several numerical techniques for solving different types of differential and integral equations, in [ 26 ] the authors introduce the solution the ABC- fractional Riccati and Bernoulli equations by using RKHS method. To explain the important of (RKF), (RKHS) you can read more details from Reproducing kernel for solving mixed type singular time-fractional partial integrodifferential equations [ 27 ], singularly perturbed boundary value problems with a delay [ 28 ], strongly nonlinear Duffing oscillators [ 29 ], integrodifferential systems with two-points periodic boundary conditions [ 30 ], Bagley–Torvik and Painlevé equations of fractional order [ 31 ], fuzzy fractional differential equations in presence of the Atangana–Baleanu–Caputo differential operators [ 32 ], time-fractional Tricomi and Keldysh equations [ 33 ], ABC–Fractional Volterra integro-differential equations [ 34 ], the Atangana–Baleanu fractional Van der Pol damping model [ 35 ], time-fractional partial differential equations subject to Neumann boundary conditions [ 36 ], singular integral equation with cosecant kernel [ 37 ]. In this paper, a new numerical method is proposed for solving 1-D interface problems of fractional order.…”