A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Abstract

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

Extension of Cubic B-Spline for Solving the Time-Fractional Allen–Cahn Equation in the Context of Mathematical Physics

“…Over the past few years, numerical equations have become a powerful and useful mathematical tool for studying many phenomena in science and engineering. The study of differential equations is multidisciplinary and is applied in many ways including control, flexibility, circuit systems, heat transfer, quantum mechanics, fluid mechanics, biomathematics, biomedicine systems, traffic turbulence, complex systems and pollution control etc [1][2][3][4]. The ACE is a simple model of a nonlinear reaction-diffusion process.…”

On Multiple-Type Wave Solutions for the Nonlinear Coupled Time-Fractional Schrödinger Model

“…They are essential for understanding and predicting the behavior of complex systems in both natural and engineered environments. However, due to the inherent complexity of NLPDEs [1][2][3][4][5][6], finding exact solutions using a single technique is often challenging. To address this, several reliable methods have been proposed, for instance, the modified exp(−ϕ(ω))expansion function [7,8], the sin-Gordon-expansion [9], the G ′ G 2 -expansion function [10], the first integral approach [11], and the Hirota bilinear approach [12,13].…”