A SIXTH-ORDER HERMITE COLLOCATION TECHNIQUE FOR NUMERICAL STUDY OF NONLINEAR FISHER AND BURGERS–FISHER EQUATIONS

Abstract This study aims to formulate a highly accurate numerical method, specifically a seventh-order Hermite technique with an error term of sixth order, to solve the Fisher and Burgers–Fisher equations. This technique employs a combination of orthogonal collocation on the finite element method and hepta Hermite basis functions. By ensuring continuity of the dependent variable and its first three derivatives across the entire solution domain, it achieves a remarkable level of accuracy and smoothness. The space discretization is handled through the application of hepta Hermite polynomials, while the time discretization is managed by the Crank–Nicholson scheme. The stability and convergence analysis of the scheme are discussed in detail. To validate the accuracy of the proposed technique, three examples are taken. The results obtained from these examples are thoroughly analysed and compared against the exact solutions and reliable data from the existing literature. It is established that the proposed technique is easy to implement and gives better results as compared with existing ones.