Analytical solution of PDEs

Part of the CFD lecture set.

See also Courant Hilbert I: Section 5

Introduction

Most methods for analytically solving PDEs transform them into systems of ODEs (ordinary differential equations). A fairly comprehensive list of techniques might include:

• Separation of variables - reduces a PDE of $ n $ independent variables into $ n $ ODEs
• Integral transforms - reduce a PDE of $ n $ variables into a PDE of $ n-1 $ variables (so, useful for 2-variable PDEs)
• Integral equations - changes a PDE into an integral equation, solved using other techniques
• Change of coordinates - changes a PDE into an ODE (or, into an easier PDE) by changing the problem coordinates
• Dependent variable transforms - transform the PDE unknown into a new, easier-to-find unknown
• Perturbation methods - changes a nonlinear problem into a sequence of linear problems that can approximate the nonlinear problem
• Impulse-response technique - decomposes initial and boundary conditions of the problem into simple impulses to find the response to each impulse; the final response is the sum of the simple impulse responses
• Calculus of variations - reformulates equations into minimization problems, e.g. total energy minimization
• Eigenfunction expansion - finds solutions of the PDE in the form of infinite sums of eigenfunctions

Separation of Variables

Courant Hilbert II:

Section 3 Part 1 (p.40 of PDF)