Method of Separation of Variables for Heat Equation - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Solution to Wave Equation by Traveling Waves

We will use the method employed by [1] to solve the wave equation

on the real line given initial position  The differential equation can be factored

Using the method of characteristic coordinates we reduce the PDE to a simpler ordinary differential equation that we are able to solve. We dene the characteristic coordinates

 and a new function  As a consequence of the chain rule,

This is a simple problem that can be solved by integrating twice to obtain

and nally changing back to the original variables wefind

The solution is the sum of two traveling waves, F and G, moving in opposite directions. Using the initial conditions we can write the sum of the two traveling waves as a function of the initial position function and the initial velocity function.

We can solve for F (x) and G(x)

where d; e are constants of integration. If we use the initial condition φ(x) = u(x; 0) we can solve for the constants and nd d + e = 0. We conclude that the solution is

Solution To Wave Equation by Superposition of Standing Waves (Using Separation of Variables and Eigenfunction Expansion)

The wave equation on a nite interval can also be solved by the superposition of standing waves as shown in [2]. We consider standing waves on a string xed at both ends u(0, t) = u(l, t) = 0, with initial velocity u_t (x, 0) = ψ(x). For simplicity we normalize velocity appropriately so that c = 1. In order to solve the equation we write u as the product of two functions, each of one variable only, to make the problem simpler to solve. Then we can use the linearity of the solution to sum over all of the solutions in order to nd the general solution. Assuming u can be written as the product of one function of time only, f (t) and another of position only, g(x), then we can write u(x, t) = f (t)g(x). We plug this guess into the differential wave equation

In order for this equation to hold, both fractions must equal some constant, λ, that neither depends on t nor on x. We can then set the equations equal to zero and try to nd solutions.

We will only consider values of λ < 0 because these are the only values that cause u to exhibit wave behavior. If λ = 0 then u(x, t) will be linear and we will not have oscillation. If λ > 0 then u will increase exponentially to innity which from a physical standpoint does not make sense in our problem. This reduces the problem to two ordinary di
erential equations that can be solved by linear combinations of trigonometric functions. If we dene λ = -m² then we know solutions exist of the form

The boundary condition g(0) = g(l) = 0 implies C = 0 and  A linear combination of solutions is also a solution so the most general solution must be

If we are given the initial position of the string u(x, 0) = φ(x) then

and we can solve for Am if we multiply both sides by sin  and integrating from 0 to l

We have used the fact

Since each term of the sum is zero when when n = m wefind

We can use a similar method to solve for B_m. We differentiate u with respect to time and use the same integration technique that was used to solve for A_m.

We have written a function de ned on the interval [0; l] as an in nite trigonometric series. It is not clear a priori if this series converges, or what limitations we must place on the function to ensure convergence of the series. The function represents the initial shape of a string and from a physical perspective this places some very strong restrictions on the function, φ(x).

We ignore the question of convergence of the in nite trigonometric series in equation (6.7) for the moment and let l = π as a simpli cation. If φ(x) is an odd function, then we can extend the function to the interval [-π, π] by instead summing from n = -∞ to n = ∞. If f (x) were an even function we would hope to be able to use the exact same technique and replace the sine terms with cosine terms. Since we can write any function as the sum of an odd and even function, by applying Euler's identity we hope to be able to express any function in this manner

We can solve for the coeffcients by multiplying both sides by e^-imx and integrating to get

Formally we expect to write each coeffcient a_m as

The series in equation (6.14) only converges if we require φ(x) to be an element of a particular space of functions. However, it is beyond the scope of this paper to delve further into this subject.