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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]

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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]

The solution to a linear Volterra integral equation of the second kind, given by the equation:[3] y ( t ) = g ( t ) + ( V y ) ( t ) {\displaystyle y(t)=g(t)+({\mathcal {V}}y)(t)} can be described by the following uniqueness and existence theorem.[3]

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.

where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernel K(x, y) and the eigenfunction tag_hash_113(y). (The limits on the integral are fixed, analogously to the limits on the sum over j.) This gives a linear homogeneous Fredholm equation of the second type.

In general, K(x, y) can be a distribution, rather than a function in the strict sense. If the distribution K has support only at the point x = y, then the integral equation reduces to a differential eigenfunction equation.

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.

Integral Equations and Operator Theory focuses on publishing original research papers in operator theory and in areas where operator theory plays a key role, such as integral equations and differential equations.Welcomes new significant developments in operator theory.Emphasizes the linear aspects of the theory.Includes fruitful interactions of operator theory with applications and other areas of mathematics.Publishes high quality research papers as well as challenging open problems.Features a very high percentage of authors intending to publish in the journal again.

In this paper, we prove the convergence of the method of successive approximations used to approximate the solution of two-dimensional nonlinear fuzzy Fredholm integral equations of the second kind. Also, we present an iterative procedure based on ...

We study the numerical solution procedure for two-dimensional Laplace's equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical ...

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this, we introduce the concept of "blow-up collocation solution" and analyze numerically some blow-up ...

The goal is to find (or prove the existence of) real functions $f$ and $g$ satisfying the above equations.I have gathered that this would be called a "nonlinear system of homogeneous Fredholm equations of the first kind."

I have somewhat limited experience with integral equations. All of my initial attempts at making sense of the system have come up short. At this point, some of my questions (I would appreciate insights into any of them!)

Differential and Integral Equations publishes carefully selected research papers on mathematical aspects of differential and integral equations and on applications of the mathematical theory to issues arising in the sciences and in engineering.

I see a lot of these laws expressed as differential equations first, and then the integral equivalent are formulated (e.g. for computational purposes, the Navier-Stokes expressed in integral form to handle discontinuities like compressible flows with shocks)

We consider the linear-impact case in the continuous-time market impact model with transient price impact proposed by Gatheral (2008). In this model, the absence of price manipulation in the sense of Huberman and Stanzl (2004) can easily be characterized by means of Bochner's theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure-valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko (2009) on the nonexistence of transaction-triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.

As standard microelectronic technology approaches fundamental limitations in speed and power consumption, novel computing strategies are strongly needed. Analogue optical computing enables the processing of large amounts of data at a negligible energy cost and high speeds. Based on these principles, ultrathin optical metasurfaces have been recently explored to process large images in real time, in particular for edge detection. By incorporating feedback, it has also recently been shown that metamaterials can be tailored to solve complex mathematical problems in the analogue domain, although these efforts have so far been limited to guided-wave systems and bulky set-ups. Here, we present an ultrathin Si metasurface-based platform for analogue computing that is able to solve Fredholm integral equations of the second kind using free-space visible radiation. A Si-based metagrating was inverse-designed to implement the scattering matrix synthesizing a prescribed kernel corresponding to the mathematical problem of interest. Next, a semitransparent mirror was incorporated into the sample to provide adequate feedback and thus perform the required Neumann series, solving the corresponding equation in the analogue domain at the speed of light. Visible wavelength operation enables a highly compact, ultrathin device that can be interrogated from free space, implying high processing speeds and the possibility of on-chip integration.

I'm trying to solve the integral equation to see whether f(u) is predicted as cos(2u). Since we know that the solution to the integral equation is cos(2u), we can approximate the integral from 0 to infinity to the limits 0 to say, 5 if we make the value of the integral from 5 to infinity negligible, and this can be done by choosing t to be small. I have chosen 100 evaluation points for the integral between 0 to 5, and this implies that i am solving for 100 values of f(u). Since i need to solve for 100 values of f(u), I need to generate 100 equations, and thus need 100 values of time t. I choose 100 values for time t between 1 and 1.3 since this will ensure that the integral is negligible for values of 5 and beyond. The following is the scipy code for doing this:

Googling for "Tikhonov regularization" should get you started on how people work around issues like this. Solving integral equations is a mature field in mathematics, so googling should help you a lot here.

A quick regularization is replacing linalg.inv(A) withlinalg.pinv(A, 1e-8)This gives something more cosine looking. The magic value 1e-8 depends on the integral kernel, but when things are about rounding error, good values to try are around sqrt(finfo(float).eps) which is saying that you trust half of the ~15 digits that floating point numbers have.

Just out of curiosity, what's than the matter with integral equations? I was under the impression that they are much much difficult than "normal" differential equations. Possibly I am misunderstanding the real meaning of "integral equation".

Integral equations sure are much more difficult to deal with compared to differential equation. But there are very simple differential equations which can simply be solved by a single integration and so we also have easier integral equations which can be turned into a DE by differentiation. 006ab0faaa