By Forsberg, Johansson

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**Example text**

39). 27)). 6 SUMMARY AND CONCLUSIONS In this chapter we have examined the one-dimensional heat equation. This is a prototype example of a diffusion equation and an understanding of it will be of benefit when we discuss more general equations. The focus in this chapter is on giving an overview of a number of analytical methods that allow us to produce an exact solution to the heat equation. The techniques are: r Separation of variables r Eigenfunction expansions r Laplace transform r Fourier transform.

Define the domain Q = I × (0, T ). Let U be a vector of length n and define partitions of U as follows U I = t (u 1 , . . 36) U II = t (u l+1 , . . , u n ) We now consider the initial boundary value problem. 39). , 2000). In general, we solve a problem by finding the solution of an integral equation. To begin with, we consider a function of one variable only. The two main categories are r Fredholm integral equations r Volterra integral equations. Let f (t) be the unknown function and suppose that g(t) and K (s, t) are known functions.

This interval may be bounded, infinite or semi-infinite. Volterra integral equations are slightly different. The interval of integration is variable. 43) a The main difference between Volterra and Fredholm equations is in the limits of integration in the integral terms. We can combine PDEs and integral equations to form an integro-parabolic equation (also known as partial integro-differential equations, PIDEs). 17). 45) 0 In this equation the constants a and b associated with the various physical parameters are both positive or both negative, depending on whether the temperature feedback is negative or positive.

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