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

Restrictions:

• Linear PDEs
• Homogeneous PDEs
• Boundary conditions of the form $ \alpha u_{x}(0,t) + \beta u (0,t) = 0 $ and $ \gamma u_{x}(1,t) + \delta u(1,t) = 0 $, where $ \alpha, \beta, \gamma, \delta $ are constants (making the boundary conditions linear and homogeneous).

The principle behind separation of variables is to find solutions of the form:

$ u(x,t) = X(x) T(t) $

In principle, an infinite number of solutions exist which will satisfy the PDE and also satisfy the boundary conditions. These solutions are used as building blocks, and the sum of all of these solutions gives the total solution $ u(x,t) $.

Combination of Variables

Courant Hilbert II:

Section 3 Part 1 (p.40 of PDF)