Integral equations arise in various physical and engineering problems. The methods of solving integral equations can be broadly classified into two categories: analytical and numerical approach.Analytical approaches use the power of mathematical analysis and differential equations to find approximate solutions. These include: Laplace transforms, Fourier transforms, complex variable methods such as conformal mapping, modeling using delay differential equation (DDE) or wave equation (WD), etc.Numerical approach uses approximation techniques such as brute-force (iterative), shooting (or falling) method, Gaussian quadrature, the trapezoidal rule etc., to numerically compute solutions in many cases when analytical approaches are not possible or feasible to use. The aim of the present chapter is to present the "brute-force" integration method, also known as "standard method" or "direct integration". The reader should be aware that the brute-force integration technique requires essentially more computer time than any other approach. Thus it is used only for special cases where analytical or other methods are not suitable.Basic concept: The concept of integral equation arises when we try to find the relationship between two functions formula_1 and formula_2 (sometimes denoted as formula_3 and formula_4 respectively), which are defined over a region formula_5 in space (or over an interval in time). When the functions are defined over the same region (or interval in time), they will be related by an integral equation. The most common examples involve the evaluation of elementary functions defined over an interval in time, such as solving equation formula_6 for formula_7, or evaluating formula_8 at some certain point in space.This is certainly not restricted to elementary functions only. Consider for instance, the problem of finding the total volume of liquid contained in container "C", which is filled exactly up to brim with liquid. Such liquid fills container C, with liquid level linearly decreasing from top to bottom, so that volume at any point in time is equal to sum volume of liquid at all earlier and later times. If we denote by formula_9 the volume of liquid in container at time formula_10, then this equation represents an integral equation.Notation: We will consider system of differential equations … Using this notation, integral equation can be written as where formula_1 is a function of space, formula_2 is a function of time.Possible types of integral equations are the following:

The following statements are true for any region "R" in three-dimensional space:

The last statement above can be demonstrated by considering areas under curves given in Figure 4.1. The area of triangle ABC is equal to its base BC multiplied by its height AC, or formula_23. This is different from the area of triangle BCD, which would be the sum of base times height.We can therefore say that integral equation gives information about the behavior of an undetermined quantity over an interval in space.To illustrate how integral equation can be applied, consider a simple example where formula_24 is a function of time given by For this example, if formula_26 is any function defined on all of "R", then any integral equation formula_27 about "R" yields an undetermined function over "R".

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2021.11

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