Topics in Differential Equations and Integral Transforms

The document contains the contents.

We know that linear differential equations with constant coefficients can be solved and their solutions are elementary functions.* There also exists some methods in which linear differential equations (with some specific types of variable coefficients) can be solved and their solutions are elementary functions. In general, a linear differential equation with variable coefficients may not admit of any solution which is expressible in terms of elementary functions. In the present chapter, we shall learn the method of solving linear differential equations and obtaining solutions in the form of an infinite series.

With the knowledge of series method of solving second order differential equations, we are ready to solve three very important differential equations of Physics. The present chapter and the next two chapters would be devoted to these differential equations. In this chapter, we shall consider the differential equation known as Legendre equation*.This differential equation occurs particularly in boundary value problems for spheres.

In this chapter, we shall study a very important differential equation called Bessel equation*. This differential equation appears in problems relating to electric fields, viberations, heat conduction, etc.

In this chapter, we shall study the third very important differential equation used very widely in applied mathematics. This differential equation is called Gauss hypergeometric equation* or simply hypergeometric equation.

In the present chapter, we shall study the solution of a special type of second order differential equations with given conditions. This type of differential equations with given conditions are called Sturm-Liouville problems.* These problems will introduce us to the concepts of ‘eigen value’ and ‘eigen function’.

Partial differential equations arise in applied mathematics and mathematical physics when the functions involved depend on two or more independent variables. The use of partial differential equation is enormous as compared to that of ordinary differential equations. In the present chapter, we shall learn the method of solving various types of partial differential equations.

In the last chapter, we studied the methods of forming partial differential equations. The next step is to solve partial differential equations. Solving a partial differential equation means to find a function which satisfies the given partial differential equation. A function satisfying a partial differential equation is called its solution (or integral). In the present chapter, we shall confine ourselves to the solution of partial differential equations of first order and at the same time linear in p and q.

By now we have learnt the method of solving first order partial differential equations which are linear in partial derivatives p and q. A partial differential equation of first order need not be linear in p and q. In the present chapter, we shall study the methods of solving such equations. In the first part, we shall study the method of solving some special types of equations which can be solved easily by methods other than the general method. In the second part, we shall take up Charpit’s general method of solution.

Till now we have been discussing the methods of solving partial differential equations of the first order. A partial differential equation of the first order involves, only the first order partial derivatives (p and q) of the dependent variable z. Now we shall consider the solution of partial differential equations of order higher than one.

From the last chapter, we have been solving linear partial differential equations with constant coefficients. In that chapter we found the general solution of only such equations in which the orders of all partial derivatives involved in the equation were same. In other words, we solved only homogeneous linear partial differential equations with constant coefficients. In the present chapter, we shall learn the methods of finding general solution of linear partial differential equations which are not homogeneous

Till now we have been discussing the solution of linear partial differential equations which are with constant coefficients. Now we shall consider the method of solving a particular type of linear partial differential equations with variable coefficients that are capable of reducing to a linear partial differential equations with constant coefficients

In the last chapter we discussed the methods of solving some special type of linear partial differential equations with variable coefficients which were capable of being reduced to linear partial differential equations with constant coefficients by changing the independent variables. Solving any given partial differential equation with variable coefficients is not an easy task. We are moving in this direction step by step.

The Laplace transform* of a suitably defined function f of a real variable t is a related function F of a real variable s. The use of Laplace transforms provide a powerful method of solving differential and integral equations. The Laplace transform method also has the advantage that it solves initial value problems directly without first finding a general solution. The ready tables of Laplace transforms has reduced the problem of solving differential equations to merely algebraic manipulation.

We have already noted that the main purpose of studying Laplace transforms is for solving various types of differential equations. During the process of solving a differential equation, we shall also require to find a function when its Laplace transform is known. This is the reverse process of finding the Laplace transform of a function. In the present chapter, we shall learn to find the function whose Laplace transform is known.

Integral equations occur very frequently in the fields of mechanics and mathematical physics. The development of the theory of integral equations began with the works of Italian mathematician V. Volterra (1896) and the Swedish mathematician I. Fredholm (1900).

In the present chapter, we shall study the method of solving a system of differential equations using Laplace transformations of functions. Our knowledge of finding Laplace transforms and inverse Laplace transforms would help us to solve the systems of differential equations. In the method of Laplace transforms, we would be able to find the required particular solution of the given system of differential equations with known initial conditions, without the necessity of first finding the general solution and then evaluating the arbitrary constants by using the given conditions.

Fourier* transforms play an important part in the theory of many branches of science. The use of Fourier transforms is indispensible in solving the problems in the field of mathematics, physics and engineering. Many types of integrals and differential equations can be solved by using Fourier transforms

When the function involved in a physical (or geometrical) problem involves more than one independent variable, a partial differential equation is obtained. The solution of this partial differential equation gives the required function. Fourier transforms are a very important tool in solving such partial differential equations.