Linear Integral Equation
Clara Vanliere
In mathematics, integral equations are equations in which an unknown function appears under an integral sign.[1] In mathematical notation, integral equations may thus be expressed as being of the form: f ( x 1 , x 2 , x 3 , . . . , x n ; u ( x 1 , x 2 , x 3 , . . . , x n ) ; I 1 ( u ) , I 2 ( u ) , I 3 ( u ) , . . . , I m ( u ) ) = 0 \displaystyle f(x_1,x_2,x_3,...,x_n;u(x_1,x_2,x_3,...,x_n);I^1(u),I^2(u),I^3(u),...,I^m(u))=0 where I i ( u ) \displaystyle I^i(u) is an integral operator acting on u.[1] Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.[1] A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows: f ( x 1 , x 2 , x 3 , . . . , x n ; u ( x 1 , x 2 , x 3 , . . . , x n ) ; D 1 ( u ) , D 2 ( u ) , D 3 ( u ) , . . . , D m ( u ) ) = 0 \displaystyle f(x_1,x_2,x_3,...,x_n;u(x_1,x_2,x_3,...,x_n);D^1(u),D^2(u),D^3(u),...,D^m(u))=0 where D i ( u ) \displaystyle D^i(u) may be viewed as a differential operator of order i.[1] Due to this close connection between differential and integral equations, one can often convert between the two.[1] For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation.[1] In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form.[2] See also, for example, Green's function and Fredholm theory.
Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogenous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations.[1] These distinctions usually rest on some fundamental property such as the consideration of the linearity of the equation or the homogeneity of the equation.[1] These comments are made concrete through the following definitions and examples:
Fredholm: An integral equation is called a Fredholm integral equation if both of the limits of integration in all integrals are fixed and constant.[1] An example would be that the integral is taken over a fixed subset of R n \displaystyle \mathbb R ^n .[3] Hence, the following two examples are Fredholm equations:[1]
Volterra: An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable.[1] Hence, the integral is taken over a domain varying with the variable of integration.[3] Examples of Volterra equations would be:[1]
Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is an improper integral.[7] This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated.[1]
In the following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations for doing so, among them being that integral equations can often be more readily solvable and are more suitable for proving existence and uniqueness theorems.[7]
It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the electric-field integral equation (EFIE) or magnetic-field integral equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem.
Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations.
This problem comes up in an exotic problem from auction theory where I want to find the optimal bidding function B[v] for v in [0,1]. Is there any way to get a solution to this problem in Mathematica? Closed-form solutions are preferred (duh!) but a plot of a numerical one is also OK.
This will involve two approximations. First, we will approximate the function B[x] by its values at n particular points in the range x, 0, 1. The integral over x will be replaced by a weighted sum over n, i.e., a quadrature rule. Second, we will only exactly satisfy the integral equation at those n points. It will hopefully be approximately satisfied at other points.
The first part abscissae is the list of n points in the range 0, 1 at which we will approximate the solution. The second part weights is the vector of weights for function values at those points to compute an integral estimate.
We will find a root using FindRoot, which likes to have a decent guess for the values of the c[i] to start with. By experimenting a little I learned that a linear solution of the form B[x]==0.2*x is a decent guess:
The solution is not exactly correct. In fact, for very small values of x, the residual (the amount by which the integral equation is not satisfied) is substantial. Plot the residual for a few values of x:
For the benefit of people solving related problems, let me just mention that we used FindRoot to search for a root (starting from a plausible guess) because this is a nonlinear integral equation. For a linear integral equation, you can use the same collocation method, but the integralEquations will be linear, so you can be sure of finding a solution simply by using Solve, or reformulate as a matrix problem and use LinearSolve.
This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter.
"This is a good introductory text book on linear integral equations. It contains almost all the topics necessary for a student. The presentation of the subject matter is lucid, clear and in the proper modern framework without being too abstract." (ZbMath, 1999)
I'm having trouble solving an integral equation. It appears to me to be a homogeneous Fredholm equation of the second kind. However, I'm being told that this can't be a Fredholm equation, because it is non-linear. Could someone help me in trying to figure out how to classify an integral equation as linear or non-linear. Also, I'll post the equation I need to solve below, and it would be great if anyone could also give me some tips on how to try and solve it. Thank you to all who reply.
An important observation is that the kernel has a singularity in the integration domain, for $0 \le x \le 1$, which makes the equation a singular Fredholm integral equation of the second kind.
An equation containing the unknown function under the integral sign. Integral equations can be divided into two main classes: linear and non-linear integral equations (cf. alsoLinear integral equation;Non-linear integral equation).
For simplicity, only integral equations in the one-dimensional case will be considered, when $D$ is a finite interval $[a,b]$. In this case, linear equations of the first and second kind can be represented in the following form:
A Fredholm kernel need not have eigen values (for example, in the case of aVolterra kernel, see below). If the kernel is symmetric and does not vanish almost-everywhere, then it has at least one eigen value and all its eigen values are real.
$$\phi(x)-\int_a^x K(x,s)\phi(s)ds = f(x), \quad a\le s\le x\le b,\label8$$These equations are called Volterra equations of the first and second kind, respectively (cf.Volterra equation). Special cases of integral equations began to appear in the first half of the 19th century. Integral equations became the object of special attention of mathematicians after the solution of theDirichlet problem for theLaplace equation had been reduced to the study of a linear integral equation of the second kind. The construction of a general theory of linear integral equations was begun at the end of the 19th century. The founders of this theory are considered to be V. Volterra (1896), E. Fredholm (1903,[Fr]), D. Hilbert (1912,[Hi]), and E. Schmidt (1907, ). Even before these investigations, the method of successive approximation for the construction of a solution of an integral equation was proposed (cf. alsoSequential approximation, method of). This method was initially applied to the solution of non-linear equations of Volterra type (in modern terminology) in connection with studies of ordinary differential equations in the work of J. Liouville (1838), L. Fuchs (1870), G. Peano (1888), and others; as well as by C. Neumann (1877) in constructing a solution of an integral equation of the second kind. The general form of the method of successive approximation is due to E. Picard (1893).
Hilbert (1904) showed that the Fredholm theorems can be proved by a rigorous application of the process of limit transition and constructed a general theory of linear equations on the basis of the theory of linear and bilinear forms in an infinite number of variables. Schmidt
gave a simpler and somewhat more general version of the investigations of Hilbert. He constructed a theory of linear integral equations with real symmetric kernel (cf.Integral equation with symmetric kernel) independently of the Fredholm theory by representing the kernel as the sum of a degenerate and a "small" kernel. T. Carleman achieved a substantial weakening of the restrictions imposed on the data and the unknown elements in the theory of integral equations of the second kind for the case of real symmetric kernels. He extended the method of Fredholm (see[Ca]) to the case when the kernel of (3) satisfies condition (4). In papers of F. Riesz (1918) and J. Schauder (1930), Fredholm's theorems were generalized to a certain class of linear operator equations in Banach spaces.