Separation of Variable Method

This method is used to convert the partial differential equations into ordinary differentialequations. It involves a solution, which splits up into a product of functions, where eachone contains only one variable. The steps to solve the differential equation are as follows:}} [1] [2] [3]

One-Dimensional Wave Equation

Let us consider the 1D wave equation (Kreysig 2010) with the boundary conditions as

where c2 = T/p. Here, T and p are the tension and mass of the deflected string.

Boundary Conditions

Here, u(x, t) is the deflection of the string, and it is fixed at the ends x = 0 and x = L. Hence, we have

Denoting the initial deflection by the functionf(x) and the initial velocity by g(x), we obtain and

We determine the wave Equation 5.1 of the following form:

which is a product of two functions F and G depending on the variables x and t, respectively. By differentiating Equation 5.5, we get

and

where

(•) denotes the derivative with respect to t primes denotes the derivative with respect to x

Using Equations 5.6 and 5.7, Equation 5.1 becomes

Equation 5.8 is transformed into Then, Equation 5.9 may be written as where k is the arbitrary constant.

Equation 5.10 will give the following two ordinary linear differential equations, that is: and

Choosing k = - p2, the general solution of Equation 5.11 becomes We know

If G = 0, then u = 0 which has no interest. Hence, G Ф 0.

If F = 0, then we have the following:

Equations 5.14 and 5.13 give Again, we have

We must take B Ф 0, otherwise F=0. Hence, sin pL = 0. Thus, where n is an integer.

Taking B = 1, we obtain infinitely many solutions F(x) = Fn(x), where

Solving Equation 5.12, we get where

The general solution is

Hence, Equation 5.1 may be written as

The entire solution is We have the initial condition

Using the Fourier sine series of f(x), we can write Cn as Similarly, using the condition (5.4), we obtain

Using the Fourier series, we get

  • [1] By applying this method, we shall obtain two or more ordinary differential equations depending on the governing differential equations.
  • [2] We shall determine solutions of those ordinary differential equations that satisfythe boundary conditions provided with the problems.
  • [3] Using the Fourier series, we shall compose those solutions of the ordinary differential equations and get the final solution of the governing differential equations.
 
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