2011 | OriginalPaper | Chapter

# Numerical solutions of partial differential equations

Derive the finite difference formulas for the first partial derivatives of a function of two real variables and construct the central finite difference formula to represent a first-order partial differential equation Draw a rectangular grid of points overlaid on the domain of a function of two real variables and evaluate the function at the boundary grid points Construct the computational molecule for a first-order partial differential equation in two real variables and use the molecule to evaluate the solutions to the equation at the grid points interior to the boundary Describe the solution as a set of simultaneous linear equations and use matrices to represent them Invert the coefficient matrix and thereby represent the solution to the partial differential equation as a column matrix Take account of a boundary condition in the form of the derivative normal to the boundary Obtain the central finite difference formulas for the second derivatives of a function of two real variables and construct finite difference formulas for second-order partial differential equations Use the forward difference formula for the first time derivatives in partial differential equations involving time and distance Use the Crank-Nicolson procedure for a partial differential equation involving a first time derivative Appreciate the use of dimensional analysis in the conversion of a partial differential equation modelling a physical system into a dimensionless equation