Projection methods for the solution of one class of singular integrodifferential equations

In this paper we propose a computational scheme of the general projection method for the solution of singular integrodifferential equations in the theory of stream lines and thermal conductivity. We theoretically substantiate this scheme from the standpoint of the theory of approximate methods of functional analysis. We consider particular cases of the general projection method, namely, the method of moments and the collocation method.     

Callocation methods for volterra integrodifferential equations with singular kernels

A collocation method which uses Hermite cubic elements is proposed for the solution of Volterra integrodifferential equations with singular kernels. Optimum error estimates in the uniform norm are obtained by means of interpolation operators. We also report on results of numerical comparisons with one well established method and another new “modified collocation” scheme.     

Existence and uniqueness of mild solution for fractional integrodifferential equations of neutral type with nonlocal conditions

In this paper, we prove the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations of neutral type with nonlocal conditions in a Banach space. New results are obtained by fixed point theorem.     

Natural Continuous Extensions of Runge-Kutta methods for Volterra integrodifferential equations

Summary In this paper we deal with a very general class of Runge-Kutta methods for the numerical solution of Volterra integrodifferential equations. Our main contribution is the development of the theory of Natural Continuous Extensions (NCEs), i.e. piecewice polynomial functions which interpolate the values given by the RK-method at the mesh points. The particular features of these NCEs allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations.     

Collocation-approximation methods for nonlinear singular integrodifferential equations in Banach spaces

A new approximation method is proposed for the numerical evaluation of the nonlinear singular integrodifferential equations defined in Banach spaces. The collocation approximation method is therefore applied to the numerical solution of such type of nonlinear equations, by using a system of Chebyshev functions.Through the application of the collocation method is investigated the existence of solutions of the system of non-linear equations used for the approximation of the nonlinear singular integrodifferential equations, which are defined in a complete normed space, i.e., a Banach space.     

Existence and uniqueness of Sobolev type integrodifferential equations

An imbedding method for nonlinear Fredholm integral equations gives rise to a Sobolev type integrodifferential equation. For such equations, sufficient conditions are given to guarantee local existence and uniqueness of solutions. A Picard type theorem utilizing a Lipschitz condition is obtained.     

Numerical solution to systems of delay integrodifferential algebraic equations

The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.     

Galerkin methods with splines for singular integral equations over (0, 1)

Summary In this paper a convergence analysis of Galerkin methods with splines for strongly elliptic singular integral equations over the interval (0, 1) is given. As trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions, multiplied by a weight function. Using inequalities of Gårding type for singular integral operators in weightedL ² spaces and the complete asymptotics of solutions at the endpoints, we provide error estimates in certain Sobolev norms.     

Finite-dimensional approximations of singular integrals and direct methods of solution of singular integral and integrodifferential equations

A brief survey is given of papers reviewed in RZh Matematika in the last 15 years (1965–1979) on applied methods of calculating various classes of one-dimensional and multidimensional singular integrals and quadrature methods of solving various classes of singular linear equations with integrals in the sense of principal values.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 18, pp. 251–307, 1980.     

Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations

In this paper, existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations is established via Banach fixed-point analysis approach and using the fuzzy number whose values are normal, convex, upper semicontinuous and compactly supported interval in E_N.     

Stability analysis of multilag and modified multilag methods for Volterra integrodifferential equations

Stability analysis of multilag and modified multilag methodsfor Volterra integrodifferential equations is presented, withrespect to the nonconvolution test equation where , , µ, and are real parameters. The applicationof these methods to this test equation leads to difference equationswith variable coefficients which are of Poincar type. Usingthe extension of the Perron theorem, the conditions under whichthe solutions to such equations are bounded are derived. Asa consequence, a complete characterization of stability regionsof multilag and modified multilag methods with respect to theabove nonconvolution test equation is obtained.     

A numerical solution of singular integrodifferential equations with variable coefficients

A method for the numerical solution of singular integrodifferential equations is presented where the integrals are discretized by using a convenient quadrature rule. Then the problem is reduced to a system of linear algebraic equations by applying the discretized functional equation to appropriately selected collocation points. This technique constitutes an extension of an analogous method convenient for solving singular integral equations which was proposed by the authors.     

Approximate solution of singular integrodifferential equations on closed contours of integration

Translated from Issledovaniya po Prikladnoi Matematike, No. 2, pp. 3–17, 1974.