Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method

Abstract

Fourth-order B-spline collocation method has been applied for numerical study of Fisher's equation which represents several important phenomena such as biological invasions, reaction diffusion in chemical processes and neutron multiplication in nuclear reactions, etc. Results are found to be better than second-order B-spline collocation method. It is observed that when time becomes sufficiently large, local initial disturbance propagates with constant limiting speed. Proposed method is satisfactorily efficient in terms of accuracy and stability.

Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

“…Spline functions theory is very active field of approximate theory in partial differential equations. Many researchers have proposed numerical solution for the nonlinear equations including Burger equations, such as, Galerkin B-Spline-collocation method (Bryan et al, 2017), exponential cubic B-spline differential quadrature method (Korkmaz and Akmaz, 2015), trigonometric cubic B-spline differential quadrature method (Korkmaz and Akmaz, 2018), cubic Bspline collocation method (Sharifi and Rashidinia, 2016), B-spline collocation and self-adapting differential evolution (jDE) algorithm (Luo et al, 2018), fourth-order cubic B-spline collocation method (Rohila and Mittal, 2018), cubic B-spline collocation scheme (Mittal and Arora, 2011), non-polynomial spline method (Ali et al, 2015), collocation method with cubic trigonometric B-spline (Raslan et al, 2016), collocation method with quintic B-spline method (Raslan et al, 2017), generalized differential quadrature method (Mokhtari et al, 2011), exponential cubic B-spline finite element method (Ersoy and Dag, 2015), B-spline Differential Quadrature Method (Bashan et al, 2015), and the Galerkin quadratic Bspline finite element method (Kutluay and Ucar, 2013). The septic B-spline approach has been used to establish approximate solutions for several partial differential equations (Ramadan et al, 2005;El-Danaf, 2008;Soliman and Hussien, 2005;Quarteroni et al, 2007;Karakoc and Zeybek, 2016;Geyikli and Karakoc, 2011).…”

Optimal control problems with Atangana‐Baleanu fractional derivative

“…The properties and capabilities of B‐spline curves make them widely used in computer graphics, computer‐aided design, geometric modeling, and so on 37 . Numerous problems of FDEs and fractional partial differential equations are solved with the help of B‐spline polynomials 38‐43 …”

A fourth-order B-spline collocation method for nonlinear Burgers–Fisher equation

“…phenomena such as gas dynamics, fluid mechanics, traffic flow and nonlinear acoustics, is given as Various numerical and analytical solutions of this equation are available in the literature [12][13][14][15][16][17][18][19]. Various noble methods were developed to numerically solve Fisher's reaction-diffusion equation shown in the papers [20][21][22]. The combination of these two equations is commonly known as the Burgers-Fisher equation given by (6).…”