TABLE OF CONTENTS Many physical and engineering problems when formulated in mathematical language give rise to partial differential equations. Besides these, partial differential equations also play an important role in the theories of elasticity, hydraulics, and so forth.
Since the general solution of a partial differential equation in a region R contains arbitrary constants or arbitrary functions, the unique solution of a partial differential equation corresponding to a physical problem will satisfy certain other conditions at the boundary of the region R. These are known as boundary conditions. When these conditions are specified for the time t = 0, they are known as initial conditions. A partial differential equation together with boundary conditions constitutes a boundary value problem.
In the applications of ordinary linear differential equations, we first find the general solution and then determine the arbitrary constants from the initial values. But the same method is not applicable to problems involving partial differential equations. Most of the boundary value problems involving linear partial differential equations can be solved by the method of separation of variables. In this method, right from the beginning, we try to find the particular solutions of the partial differential equation which satisfy all or some of the boundary conditions and then adjust them till the remaining conditions are also satisfied. A combination of these particular solutions gives the solution of the problem.
The fourier series is a powerful aid in determining the arbitrary functions.
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