Lecture Notes in Numerical Methods of Differential Equations

Full text available

This chapter constitutes an introduction to linear multi step methods, Runge Kutta methods, and finite difference methods which are presented in the chapters that follow. The homo- geneous and non-homogeneous linear difference equations are solved by the following Mathematica modulae: The module differenceEqn finds the general solution of a linear difference equation, the module baseSolution finds the fundamental set of solutions of a homogeneous linear difference equa- tion, the module particularSolution finds a particular solution of a non-homogeneous linear difference equation by the method of variation of coefficients. The chapter ends with a set of questions.

In this chapter, the linear multi step methods and Runge Kutta methods are considered. For both classes of methods, efficient algorithms have been built and implemented in Mathemat- ica. The designed Mathematica modulae are applied to selected initial value problems for ordinary differential equations and system of equations of the first order. Also, in chapter 2, an optimal algorithm has been built and implemented in the Mathematica module solveBVP. The module finds the numerical solution u = (u1, u2, , up) of the system of equations − d2u dx2 = f(x, u), a < x < b with the boundary value conditions u(a) = a0, u(b) = b0. As exercises, the equations on rotation of a heavy string and on equilibrium of a rod are solved by the module. The chapter ends with a set of questions.

Two techniques of analysis of convergence and the global error estimates have been developed. The first technique is based on the discrete maximum principle to prove uniform con- vergence of finite difference methods for elliptic and parabolic equations. The second technique draws on spectral analysis and deals with average convergence in the discrete Hilbert’s space H. The chapter ends with a set of questions.

In chapter 4, two finite difference schemes are built for Helmholtz equation with Dirich- let’s boundary value conditions. It is proved that the first scheme is O(h2 1 + h2 2) convergent in the norm of the Hilbert’s space H and in the maximum norm. The second scheme is O(h4 1 + h4 2 ) accurate and is also convergent in the norm of the Hilbert’s space H and in the maximum norm. Both schemes are solved by the Mathematica module. In the last section, Poisson’s equation with Dirichlet’s boundary conditions is solved by the method of lines. The chapter ends with a set of questions.

In this chapter, considerable attention is paid to construction of finite difference schemes with weight 0 ≤ σ ≤ 1 for diffusion equations in one and two space variables. Efficient algorithms are built for selected values of the weight σ, like implicit scheme (σ = 1), explicit scheme (σ = 0) and Crank Nicolson scheme (σ = 1/2). The average convergence of the scheme is proved in the norm of the Hilbert’s space H. The scheme with weight is implemented in the Mathematica module heatEqn and applied to the diffusion equations with initial boundary value conditions. In the last section, the heat equation with initial boundary value conditions is solved by the method of lines. The chapter ends with a set of questions.

In chapter 6, the finite difference scheme with weight 0 ≤ σ ≤ 1 has been built for the wave equation with initial boundary value conditions. It is proved that the scheme is convergent and the global error estimate is given. The scheme with weight  is solved by the method of separation of variables. The Mathematica module waveEqn is designed and applied to initial boundary value problems for the wave equation. In the last section, the wave equation is solved by the method of lines. The chapter ends with a set of questions.

Full text available

Full text available