Joseph Fourier, French mathematician and physicist made extensive use of Fourier series in developing the theory of heat conduction. An arbitrary periodic function f(x) (satisfying certain conditions) could be represented by a series of sines and cosines, of the form 1/2 a0 +a1 cos x+b1 sin x+a2 cos 2x+b2 sin 2x+• • • Such a series with a0, an and bn given by Euler formulae (given l
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We have already seen that a periodic function can be represented by an infinite series known as Fourier series. Fourier transform deals with non periodic functions. It can be shown that the limiting form of Fourier series as the period tends to infinity is the Fourier integral and this suggests the idea of Fourier transform. This is useful in the solution of boundary value problems and also in oth
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If a physical quantity depends on more than one independent variable (for example, in one dimensional fluid flow problems, the velocity depends on the space variable x and time t) the governing differential equations are necessarily partial differential equations. Some of the fields where partial differential equations arise are fluid dynamics, electromagnetic theory, theory of heat conduction and
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We now study three linear homogeneous second order PDEs which are important in physical applications, the wave equation, the heat equation and Laplace's equation.

In the process of analysing problems in engineering, the analysts come across systems of equations which belong to different categories like, linear algebraic, nonlinear, transcendental, differential and integral equations. School level mathematics and elementary engineering mathematics include, among other topics, solution aspects of certain simple classes of equations. But, in many contexts, it
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Interpolation is the `art of reading between the lines in a table'. To explain this briey, suppose that a data consisting of (n + 1) ordered pairs of values of the form (xi; yi); i = 0; 1; 2; _ _ _ ; n; is available. Let yi be the value of an unknown function of x at x = xi. A question of practical importance here is, what is the value of y at x = ~x where ~x is none of xi; i = 0; 1; 2; _ _ _ ; n,
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One of the important applied problems of elementary calculus is that of determining relative maxima and relative minima of functions of a single variable. The results relating to this were extended to problems of maxima and minima of functions of several variables. These were further extended to problems of finding maxima and minima of functions of several variables subject to equality constraints
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Differential equations arise in the study of systems in which variables vary continuously. When the independent variable varies by taking discrete values, the systems are often characterised by difference equations. Also, most of the numerical methods of solutions of differential equations are based on finite difference methods and involve solutions of difference equations. Further, recurrence rel
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