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

or,

$ 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) $.

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 Y}{\partial y} = 0 $

or,

$ \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y} = 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.

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: Heat Equation

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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