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
• Eigenfunction expansion - finds solutions of the PDE in the form of infinite sums of eigenfunctions
• Calculus of variations - reformulates equations into minimization problems, e.g. total energy minimization

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_{n}(x,t) = X_{n}(x) T_{n}(t) $

or,

$ u_{n}(x,t) = X_{n}(x) + T_{n}(t) $

In principle, an infinite number of solutions exist which will satisfy the PDE and also satisfy the boundary conditions (hence the subscript $ n $).

These solutions are used as building blocks, and the sum of all of these solutions gives the total solution $ u(x,t) $.

The idea behind separation of variables (and the reason it only applies to homogeneous problems) is, if one has an equation of the form

$ \frac{\partial X}{\partial x} * \frac{\partial T}{\partial t} = 0 $

or,

$ \frac{\partial X}{\partial x} + \frac{\partial T}{\partial t} = 0 $

then each of the terms must be equal to a constant - otherwise the derivative of X would create a function in x that could not be canceled out by the other derivatives.

Given these infinite solutions,

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Example 1: First Derivatives

This example uses separation of variables to solve the PDE:

$ u_{x} \times u_{y} = 0 $

This is a straightforward application of separation of variables. A solution of the form $ u(x,y) = X(x) \times Y(y) $ will be assumed. Plugging this in:

$ \frac{\partial X}{\partial x} \times \frac{\partial Y}{\partial y} = 0 $

Which means that each derivative must be constant - otherwise (as mentioned above) there would be functions of $ x,y,z $ left over:

$ \frac{d X}{d x} = c_1 $

$ \frac{d Y}{d y} = c_2 $

So, the 1st order PDE has been transformed into a set of 2 ODEs. These are easy to solve, and yield $ X = c_1 x $ and $ Y = c_2 y $, so the solution is:

$ u(x,y) = ( c_1 x ) ( c_2 y ) = c_3 x y $

Which, when plugged in, satisfies the original PDE.

Example 2: Second Derivatives

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Example 3: Heat Equation

Separation of variables can be used to solve the heat equation, given the right circumstances and boundary conditions.

Consider the heat equation in a normalized domain:

$ u_{t} = \alpha^2 u_{xx} \qquad 0 < x < 1, 0 < t < \infty $

Given the (linear homogeneous) boundary conditions

$ \begin{cases} u(0,t) = 0 & 0 < t < \infty \\ u(1,t) = 0 & 0 < t < \infty \end{cases} $

This PDE describes a finite rod, with the (relative) temperature at either end fixed at zero, and some initial temperature profile.

Combination of Variables

Courant Hilbert II:

Section 3 Part 1 (p.40 of PDF)

References

http://www.exampleproblems.com/wiki/index.php/PDE:Integration_and_Separation_of_Variables