An Improvised B Spline Collocation Method for Non Linear Partial Differential Equations

Abstract

A majority of the physical and technological phenomena can be modeled in terms of differential equations. Due to their enormous applications, extensive work is going on to solve them either analytically or numerically. Depending upon the type of differential equation, along with the initial and boundary conditions, different analytic techniques can be applied. Sometimes due to the complexity of the system or the non availability of the exact solution, different numerical methods need to be adopted to find the solution. The current research deliberates upon the numerical solution of some non linear partial differential equations by keeping prominent features in mind like higher order of convergence, less computational cost, and better accuracy in results, etc. newlineFor this purpose collocation method is opted, which is a weighted residual method. The advantage of using this method is its ease of implementation as no integral needs to be performed. To apply the collocation method, some basis functions are required, for which cubic and quintic B splines are chosen. The collocation method with B spline basis functions leads to a banded matrix, e.g., with cubic B splines, there is a three band matrix and with quintic B splines, there is a five band matrix. The sparsity of the banded matrix is an advantage of using splines as opposed to full matrices, which are obtained using other polynomials as basis functions. newlineThe present work extends the standard cubic and quintic B spline collocation method by carrying out posteriori corrections in the interpolating splines and their higher order derivatives. This is accomplished by defining different end conditions and forcing the spline interpolant to satisfy the interpolation along with end conditions. With these improvements, the technique is termed as improvised cubic B spline collocation method ICSCM or improvised quintic B spline collocation method IQBSCM accordingly. These corrections result in the enhancement of the order of convergence in the spatial domain.