# Method Of Separation Of Variable ( Developed By J. Fourier

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Chapter 3
METHODOLOGY
3.1 Method of Separation of Variable (developed by J. Fourier)
The method of separation is based on the expansion of an arbitrary function in terms of the Fourier series. This method is applied by assuming that the dependent variable is a product of a number of functions and each function being a function of a single independent variable. This reduces the partial differential equation to a system of ordinary differential equations, each being a function of a single independent variable. For the transient conduction in a plain wall, the dependent variable is the solution function θ(X, F0), which is expressed in terms of θ(X, F0) = F(X)G(t), and the application of this method results in to the two ordinary differential equations, one in terms of X and the other one in F0.
Now we demonstrate the use of the method of separation of variables by applying it to the one-dimensional transient heat conduction problem given in Eqs. (1). First,
Dimensionless differential equation is given by: ( ∂²θ)/∂X²=∂θ/(∂F_0 ) eq. (1a)
Boundary conditions: (∂T(0,t))/∂x=0
And -k (∂T(L,t))/∂x=h[T_((L,t))-T_a] eq. (1b)
Dimensionless initial condition is θ(X, 0) = 1