This book presents a unified approach to computational mathematical modelling based on differential equations combining aspects of mathematics, computation and application. The backbone of the book is a general methodology for the numerical solution of differential equations based on Galerkin's method using piecewise polynomial approximation. The book is a substantial revision of the successful text Numerical Solution of Partial Differential Equations by the Finite Element Method by C. Johnson. It begins with a constructive proof of the Fundamental Theorem of Calculus that illustrates the close connection between integration and numerical quadrature and introduces basic issues in the numerical solution of differential equations including piecewise polynomial approximation and adaptive error control. After preparatory material on linear algebra and polynomial approximation, the computational methodology is developed starting with model problems taking the form of scalar linear ordinary differential equations, then proceeding through systems of linear ordinary differential equations to linear partial differential equations including the Poisson equation, the heat equation, the wave equation and convection-diffusion-absorption equations. The book includes background material on the derivation of the differential equations as mathematical models of physical phenomena and mathematical results on properties of the solutions of differential equations. The theme of error estimation and adaptive error control is developed consistently. This text is suitable for courses in mathematics, science, and engineering ranging from calculus, linear algebra, differential equations to specialized courses on computational methods for differential equations....