Question

The heat conduction equation in two space dimensions is $$ \alpha^{2}\left(u_{x x}+u_{y y}\right)=u_{t} $$ Assuming that \(u(x, y, t)=X(x) Y(y) T(t),\) find ordinary differential equations satisfied by \(X(x), Y(y),\) and \(T(t) .\)

Short answer

Based on the provided step by step solution, the ordinary differential equations for the functions X(x), Y(y), and T(t) are as follows: $$ X''(x)=-m^2 X(x) \\ Y''(y)=-n^2 Y(y) \\ T'(t)=-\alpha^2 (m^2+n^2) T(t) $$ These equations are individual ODEs that can be solved to obtain the solution to the original heat conduction equation in two space dimensions.

Step-by-step solution

Rewrite the equation using the given assumption

The given heat conduction equation is $$ \alpha^{2}\left(u_{x x}+u_{y y}\right)=u_{t} $$ Assuming that the solution can be written as \(u(x, y, t)=X(x)Y(y)T(t)\), we can now rewrite the given partial differential equation in terms of the products of these functions.

Differentiate \(u(x,y,t)\) and separate variables

Now, we want to find the second partial derivatives of \(u(x, y, t)\) with respect to \(x\) and \(y\), and the first partial derivative with respect to \(t\): $$ u_{x x} = X''(x)Y(y)T(t) \\ u_{y y} = X(x)Y''(y)T(t) \\ u_{t} = X(x)Y(y)T'(t) $$ Substituting these derivatives into the heat conduction equation, we get: $$ \alpha^{2} \left( X''(x)Y(y)T(t) + X(x)Y''(y)T(t) \right) = X(x)Y(y)T'(t) $$ Now, let's divide both sides of the equation by \(\alpha^2 X(x)Y(y)T(t)\): $$ \frac{X''(x)}{X(x)}+\frac{Y''(y)}{Y(y)}=\frac{T'(t)}{\alpha^2 T(t)} $$ As \(x\) and \(y\) are independent variables, we can separate this equation into two separate equations, one for \(X(x)\) and \(T(t)\), and the other for \(Y(y)\) and \(T(t)\).

Set up constants for the separation and solve ODEs

To separate variables, we introduce constant \(-m^2\) and constant \(-n^2\), such that $$ \frac{X''(x)}{X(x)}=-m^2 \\ \frac{Y''(y)}{Y(y)}=-n^2 $$ and $$ \frac{T'(t)}{\alpha^2 T(t)}=-m^2-n^2 $$ Now, we have three ODEs for \(X(x)\), \(Y(y)\), and \(T(t)\): $$ X''(x)=-m^2 X(x) \\ Y''(y)=-n^2 Y(y) \\ T'(t)=-\alpha^2 (m^2+n^2) T(t) $$ These are simple ODEs that can be solved individually.

Write down the solution

The solution to the heat conduction equation in two space dimensions is given by the product of solutions of \(X(x)\), \(Y(y)\), and \(T(t)\) ODEs.

Key concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are equations that involve multivariable functions and their partial derivatives. These equations are essential in describing various physical phenomena, such as heat conduction, wave propagation, and fluid flow. In the context of the heat conduction equation, we have a PDE governing the distribution of heat in a given space over time:
\[\alpha^{2}(u_{xx} + u_{yy}) = u_{t}\]
This equation indicates how the temperature function \( u(x, y, t) \) evolves. The terms \( u_{xx} \) and \( u_{yy} \) describe the rate at which heat diffuses through space, while \( u_t \) represents the rate of change of temperature with time.

Understanding PDEs requires familiarity with calculus and the ability to manipulate partial derivatives. The separation of variables is one technique used to solve PDEs, as it reduces a complex problem into simpler, ordinary differential equations (ODEs) that can be managed more easily.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) involve functions of a single variable and their derivatives. Unlike PDEs, which involve multiple variables, ODEs are simpler as they require only one independent variable.
In solving the heat conduction equation, we assume a solution in the form of product \( u(x, y, t) = X(x)Y(y)T(t) \). This assumption allows us to transform the original PDE into three respective ODEs by isolating each function:

• \(X''(x) = -m^2 X(x)\)
• \(Y''(y) = -n^2 Y(y)\)
• \(T'(t) = -\alpha^2 (m^2+n^2) T(t)\)

This approach simplifies the problem as each resulting ODE can be solved independently using standard techniques such as characteristic equations or integrating factors.
ODEs arise in numerous natural phenomena descriptions and are fundamental in fields like physics, engineering, and biology.

Separation of Variables

Separation of Variables is a mathematical method used in solving differential equations, including PDEs. This technique involves expressing a complex equation as a product of functions, each depending on a single variable. By doing so, it separates the original multi-variable problem into simpler sub-problems.
In the context of the heat conduction equation, using separation of variables:

• Assume \( u(x, y, t) = X(x)Y(y)T(t) \) where each function represents a spatial or temporal variable.
• Substitute into the original PDE and separate the equation to form distinct parts, each involving one independent variable.
• Introduce constants (like \(-m^2\) and \(-n^2\) in this case) for each partition to ensure the equation remains balanced.

Once separated, solving these simpler ODEs becomes more straightforward.
Separation of variables is an extremely powerful method as it can transform a complex, multi-dimensional problem into a set of more manageable one-dimensional problems, each solvable using standard ODE solutions.