ABSTRACT
This chapter focuses on implicit methods for solving differential equations. It presents examples of systems which include ordinary differential equations (ODEs) that are stiff in nature, partial differential equations (PDEs) that require implicit time stepping, and systems governed by differential algebraic equations (DAEs). The chapter utilizes differential equation solvers in MATLAB for solving problems of practical interest. A formal derivation of Adams-Moulton methods, including error analysis, can be more conveniently from polynomial interpolation formulae. The backward difference formula (BDF) methods work in a different way than Adams' families of ODE solvers. Numerical difference formula (NDF) is a modification of the BDF method for solving ODEs and DAEs. A narrow stability region limits the applicability of the ODE solver, as observed for Euler's explicit method. The chapter demonstrates the application of the AM-2 (trapezoidal) method to solve stiff ODE problems and also demonstrates the Crank-Nicolson method for hyperbolic PDEs.
