Numerical study of the nonlinear Cauchy diffusion problem and Newell-Whitehead equation via cubic B-spline quasi-interpolation

In this article, a numerical approximation to the solution of the Newell-Whitehead equation (NWE) and Cauchy problem of ill-posed non-linear diffusion equation have been studied. The presented scheme is obtained by using the derivative of the cubic B-spline quasi-interpolation (BSQI) to approximate the spatial derivative of the dependent variable and first order forward difference to approximate the time derivative of the dependent variable. Some numerical experiments are provided to illustrate the method. The results of numerical experiments are compared with analytical solutions. The main advantage of the scheme is that the algorithm is very simple and very easy to implement.