Numerical method using cubic B-spline for the heat and wave equation

Quintic B-spline method for integro interpolation

“…In order to analyze the errors, we give some useful operators (see [10,[14][15][16][17][18]). For a given step h and an infinitely differentiable y(x), we define…”

Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

“…The study of B-spline functions is a key element in computer-aided geometric design [32][33][34][35]. It has also attracted attention in the literature [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] to the numerical solution of various differential equations [38][39][40]. This is because they have important geometric properties and features that make them amenable to more detailed analysis.…”

“…Numerical methods based on B-spline functions of various degrees have been utilized for solving initial and boundary value problems. As examples, a cubic B-spline collocation method was used to solve a nonlinear diffusion equation subject to certain initial and Dirichlet boundary constraints [41], a finite element method based on bivariate splines has been used for solving parabolic partial differential equation [42], and the combination of finite difference approach and cubic B-spline method was applied for the solution of a one-dimensional heat equation subject to local boundary constraints [43,44]. Goh et al [45] presented a comparison of cubic B-spline and extended cubic uniform B-spline based collocation methods for solving a one-dimensional heat equation with a nonlocal initial constraint and concluded that extended cubic uniform B-spline with an appropriate value of parameters gives better results than the cubic B-spline.…”

“…Goh et al [45] presented a comparison of cubic B-spline and extended cubic uniform B-spline based collocation methods for solving a one-dimensional heat equation with a nonlocal initial constraint and concluded that extended cubic uniform B-spline with an appropriate value of parameters gives better results than the cubic B-spline. A finite difference scheme based on cubic B-spline was also used for solving the one-dimensional wave equation [43], advection-diffusion equation [44], one-dimensional coupled viscous Burgers' equation [47], system of strongly coupled reaction-diffusion equations [48], and one-dimensional hyperbolic problems [49].…”

New High Accuracy Non-polynomial Spline Group Explicit Iterative Method for Two Dimensional Elliptic Boundary Value Problems

“…Fyfe concluded that spline method is better than the usual nite-di erence method as the spline method has the exibility to get the solution at any point in the domain with more accurate results. Due to its simplicity, many researchers started considering spline as one of the approximation tool to obtain accurate numerical solutions [16][17][18][19]. Recently, Ding et al [20] and Gopal et al [21] have studied non-polynomial spline methods for the numerical solution to one-dimensional hyperbolic problem.…”