MONIC CHEBYSHEV’S FIRST DERIVATIVE PSEUDO-GALERKIN METHOD FOR SOLVING LINEAR AND NONLINEAR IBVPS: REAL-LIFE AND PHYSICAL APPLICATIONS

The computation of the solutions of differential equations (DEs) and integral equations (IEs) has noteworthy aspects due to the fact that these equations’ numerical or analytical solutions can yield applicable benefits in various realms of categories of engineering, physics, biology, and signal processing finance, among others. The pseudo-Galerkin method, as a numerical technique for approximating solutions to various types of differential and integral equations, merges the elements concerning the Galerkin method as well as the collocation methods, turning it into a hybrid mode of approach, providing versatility, efficiency, accuracy along with reduced complexity. Accordingly, this paper uses the pseudo-Galerkin method to solve linear and nonlinear differential equations depending on the monic of Chebyshev’s first derivatives as new basis functions through which new operational matrices for derivatives for the introduced basis function have been formed. In addition, a new linearization relation has been proved. Consequently, this relation and other relations have been used via the pseudo-Galerkin method to transform the given initial boundary value problems (IBVPs) into an algebraic system of equations. The error analysis study verifies the convergence of the proposed method. Finally, elements of accuracy, efficiency, and correctness are investigated by solving several numerical applications and real-life problems, including an environmental pollution model.