This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space.

This paper aims to explore the numerical solution of non-linear fractional-order Burgers’-Huxley equation based on Caputo’s formulation of fractional derivatives. The equation serves as a versatile tool for analyzing a wide range of physical, biological, and engineering systems, facilitating valuable insights into nonlinear dynamic phenomena. The fractional operator provides a comprehensive mathematical framework that effectively captures the non-locality, hereditary characteristics, and memory effects of various complex systems. The approximation of temporal differential operator is carried out through finite difference based L1 scheme, while spatial discretization is performed using modified cubic B-spline basis functions. The stability as well as convergence analysis of the approach are also presented. Additionally, some numerical test experiments are conducted to evaluate the computational efficiency of a modified fourth-order cubic B-spline (M43BS) approach. Finally, the results presented in the form of tables and graphs highlight the applicability and robustness of M43BS technique in solving fractional-order differential equations. The proposed methodology is preferred for its flexible nature, high accuracy, ease of implementation and the fact that it does not require unnecessary integration of weight functions, unlike other numerical methods such as Galerkin and spectral methods.

This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used include splines, in particular B-splines, and Hermite functions. Spline functions are useful for obtaining global approximations whilst Hermite functions are useful for approximation over finite elements. Our aim in this review is to study quintic Hermite functions and develop a numerical collocation scheme for solving ODEs and PDEs. This choice of basis functions is further motivated by the fact that we are interested in solving problems having solutions with steep gradients and oscillatory properties, for which this approximation basis seems to be a suitable choice. We derive the quintic Hermite basis and use it to formulate the orthogonal collocation on finite element (OCFE) method. We present the error analysis for third order ODEs and derive both global and nodal error bounds to illustrate the super-convergence property at the nodes. Numerical simulations using the Julia programming language are performed for both ODEs and PDEs and enhance the theoretical results.