Numerical solution of functional differential equations via modified collocation methods / Muhammad Bilal

Centre for Mathematical Sciences

Thesis (Doctor of Philosophy) -- Universiti Malaysia Pahang – 2022

Includes bibliographical reference

Mathematical model of functional differential equations (FDE) are frequently employed to represent the dynamics of natural system, in particular in biological and physical phenomena. Among all of the FDE, pantograph differential equations and fractional pantograph differential equation have attracted an increasing attention due to the capability of the models to provide meaningful description as well as can predict future behaviour under moderately changed conditions. The models incorporate time delay which is an essential feature of realistic phenomena. However, the complexity arise in solving the pantograph and fractional pantograph differential equations because of the presence of time delay and the fractional form of the equations. There is a need to propose a reliable and an efficient numerical methods for solving pantograph and fractional pantograph differential equation. The main objective of the thesis is to perform modification on collocation method by proposing new types of polynomials embedded in the methods for solving pantograph differential equations and fractional pantograph differential equations. The noteworthy feature of collocation method is the solution can be represented in matrix form, hence easy to be programmed and able to reduce the calculation time. Furthermore, collocation method can be easily generalized to solve nonlinear pantograph and fractional pantograph differential equations of high order. Lucas, Ortho exponential, Boubeker and Hermite polynomials are employed to solve pantograph differential equations. Taylor and Bessel polynomials are embedded in collocation method to solve fractional pantograph differential equation. Convergence analysis of the proposed methods are presented and the methods have been shown converge faster than reported solutions. Numerical examples are carried out and the comparison of the methods with the existing methods in literature shows that the proposed methods produce low values of error, hence indicate good performance of the proposed methods compare than the methods in literature. The errors are computed using absolute errors and residual errors. Stability analysis showed that the collocation method with polynomials Lucas, Ortho exponential, Boubeker and Hermite was stable in solving pantograph differential equations when the size of matrix is increased. Whereas, the collocation method with Taylor and Boubeker polynomials is stable in solving fractional pantograph differential equation when the matrix size is increased. This study provide guidance to researchers in the selection of numerical methods, especially the choosing of certain polynomials that are generalized with the collocation method to find the approximate solution of pantograph and fractional pantograph differential equation.