# Cordial volterra integral equations of rst kind

Two-dimensional first-kind Volterra integral equations VIEs are studied. The first-kind equations are reduced to second kind, and by obtaining an appropriate integral inequality, existence and uniqueness are demonstrated. The equivalent discrete integral inequality then permits convergence of discretization methods; and this is illustrated for the Euler method. Finally, a class of nonlinear telegraph equations is shown to be equivalent to two-dimensional Volterra integral equations, thereby providing existence and uniqueness results for this class of equation.

Furthermore, the telegraph equation may be solved by the numerical method for two-dimensional VIEs, and a simple numerical example is given. Most users should sign in with their email address.

Select Format Select format. Permissions Icon Permissions. Issue Section:.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. In this case, linear equations of the first and second kind can be represented in the following form:.

Equations of the second kind are most frequently encountered in mathematical physics. An important example of a Fredholm equation is one in which the kernel satisfies the condition. The homogeneous integral equation corresponding to equation 2 is similarly defined. Such a kernel is called symmetric.

A Fredholm kernel need not have eigen values for example, in the case of a Volterra kernelsee 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. Volterra equation. Special cases of integral equations began to appear in the first half of the 19th century.

### Volterra integral equation

Integral equations became the object of special attention of mathematicians after the solution of the Dirichlet problem for the Laplace 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. VolterraE. Fredholm[Fr]D. Hilbert[Hi]and E. Schmidt. Even before these investigations, the method of successive approximation for the construction of a solution of an integral equation was proposed cf.

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. LiouvilleL. FuchsG. Peanoand others; as well as by C.

Neumann 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 In studying the equation of a vibrating membrane H. This conjecture was proved by Fredholm — The work of Fredholm was preceded by investigations of Volterra —who studied integral equations of the form 78. Following Volterra, Fredholm replaced the integral in 3 by a Riemann integral sum and considered the integral equation 3 as a limiting case of a finite system of linear algebraic equations see Fredholm equation.

Fredholm's theory for equation 3 was extended to the case of a system of integral equations and also to the case of a kernel with weak singularity see Integral operator.

The solution of a system reduces to that of a single equation, the kernel of which has lines of discontinuity parallel to coordinate axes. Hilbert 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.

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. 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.

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In papers of F.This paper contains a study of numerical methods for solving linear Volterra integral equations of the first kind. Means for improving the results of the convergent methods are discussed.

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Abstract This paper contains a study of numerical methods for solving linear Volterra integral equations of the first kind. Issue Section:. Download all slides. View Metrics. Email alerts Article activity alert.Full-text: Access denied no subscription detected We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text. Alternatively, we obtain uniqueness in the class of locally integrable functions with locally integrable mean. We further discuss a uniqueness-of-continuation problem where the conditions on the kernel need only be satisfied in some neighborhood of the diagonal.

We illustrate with examples the necessity of the conditions on the kernel and on the uniqueness class, and sketch the application of the theory in the context of a nonlinear model. Source J. Integral Equations ApplicationsVolume 31, Number 3 Zentralblatt MATH identifier Keywords Cordial Volterra equations weakly degenerate kernel uniqueness.

Darbenas, Zymantas; Oliver, Marcel. Uniqueness of solutions for weakly degenerate cordial Volterra integral equations. Integral Equations Applications 31no. Read more about accessing full-text Buy article. Article information Source J.

Export citation. Export Cancel. References H. Brunner, Volterra integral equationsCambridge University Press Zentralblatt MATH: You have access to this content. You have partial access to this content.

You do not have access to this content. More like this.We study the approximation of solutions of a class of nonlinear Volterra integral equations VIEs of the third kind by using collocation in certain piecewise polynomial spaces. If the underlying Volterra integral operator is not compact, the solvability of the collocation equations is generally guaranteed only if special so-called modified graded meshes are employed.

It is then shown that for sufficiently regular data the collocation solutions converge to the analytical solution with the same optimal order as for VIEs with compact operators. Numerical examples are given to verify the theoretically predicted orders of convergence. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Blom, J. SIAM J.

Brunner, H. Cambridge University Press, Cambridge Google Scholar. Diogo, T. IMA J.

## Integral equation

Grandits, P. Integral Equ. Liang, H. BIT Numer. Kangro, R. Krantz, S. Lighthill, M. A— Sato, P. Seyed Allaei, S. Vainikko, G. Yang, Z.

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Download references. Part of this work was carried out during a visit of the first two authors to the Department of Mathematics at Hong Kong Baptist University. We also thank Prof.

### Analysis of collocation methods for nonlinear Volterra integral equations of the third kind

Xiao Yu for his support. The authors would like to express their gratitude to the reviewers: their valuable comments and suggestions led to a greatly improved version of the paper.

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We consider the problem of finding regularized solutions to ill-posed Volterra integral equations. The method we consider is a sequential form of Tikhonov regularization that is particularly suited to problems of Volterra type.

We prove that when this sequential regularization method is coupled with several standard discretizations of the integral equation collocation, rectangular and midpoint quadratureone obtains convergence of the method at an optimal rate with respect to noise in the data. In addition we describe a fast algorithm for the implementation of sequential Tikhonov regularization and show that for small values of the regularization parameter, the method is only slightly more expensive computationally than the numerical solution of the original unregularized integral equation. Finally, numerical results are presented to show that the performance of sequential Tikhonov regularization nearly matches that of standard Tikhonov regularization in practice but considerable savings in cost are realized.

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IVP to Volterra integral Equation example 2

Digg This. Notify Me! E-mail Alerts. RSS Feeds. SIAM J. Related Databases. Web of Science You must be logged in with an active subscription to view this. Keywords first-kind Volterra equationsinverse problemsregularizationsequential numerical methodsfast algorithm. Publication Data. Publisher: Society for Industrial and Applied Mathematics. Patricia K. Advances in Mathematical Physics Boundary Value Problems Inverse Problems 33 :8, International Journal of Thermal Sciences Advances in Difference Equations Fixed Point Theory and Graph Theory, Applied Mathematics and Computation European Journal of Operational Research :1, International Journal of Heat and Mass Transfer 62 Mathematical and Computer Modelling 57 Experimental Mechanics 52 :4, Acta Materialia 60 :5, Archive of Applied Mechanics 80 :3, Journal of Inverse and Ill-posed Problems 17 Zhewei Dai and Patricia K.

Applied Thermal Engineering 27In mathematicsthe Volterra integral equations are a special type of integral equations. A linear Volterra equation of the second kind is. In operator theoryand in Fredholm theorythe corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition methodis due to George Adomian. A linear Volterra integral equation is a convolution equation if.

Such equations can be analyzed and solved by means of Laplace transform techniques. InLalescu wrote the first book ever on integral equations. Volterra integral equations find application in demographythe study of viscoelastic materials, and in actuarial science through the renewal equation.

One area where Volterra integral equations appear is in ruin theorythe study of the risk of insolvency in actuarial science. From Wikipedia, the free encyclopedia. Handbook of Integral Equations 2nd ed. Cambridge Monographs on Applied and Computational Mathematics. School of Mathematics, Statistics and Actuarial Science. University of Kent.

February 20, Categories : Integral equations. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.