To the theory of volterra integral equations of the first kind with discontinuous kernels
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Volterra equations
Quadrature (astronomy)
Kernel (algebra)
In this paper, the solving of a class of both linear and nonlinear Volterra integral equations of the first kind is investigated. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The method of successive approximations (Neumann’s series) is applied to solve linear and nonlinear Volterra integral equation of the second kind. Some examples are presented to illustrate methods.
Neumann series
Independent equation
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Quadrature (astronomy)
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В математических моделях физических явлений, использующих результаты экспериментов, зачастую возникает необходимость решения дифференциальных уравнений.Подобные задачи относятся к классу некорректных математических задач.В данной работе для получения приближенного решения дифференциального уравнения первого порядка с определенными краевыми условиями выполняется построение соответствующего регуляризирующего алгоритма.Реализуется метод, заключающийся в построении эквивалентного исходному дифференциальному урав
Volterra equations
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This paper addresses the asymptotic behaviors of a linear Volterra type integro-differential equation. We study a singular Volterra integro equation in the limiting case of a small parameter with proper choices of the unknown functions in the equation. We show the effectiveness of the asymptotic perturbation expansions with an instructive model equation by the methods in superasymptotics. The methods used in this study are also valid to solve some other Volterra type integral equations including linear Volterra integro-differential equations, fractional integro-differential equations, and system of singular Volterra integral equations involving small (or large) parameters.
Volterra equations
Limiting
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In this paper, we study the solvability of a second-kind Volterra integral equation. By replacing the right-hand side and the unknown function, the integral equation is reduced to an integral equation, the kernel of which is not «compressiblek. Using the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. The solution of the homogeneous integral equation corresponding to the original nonhomogeneous integral equation found in explicit form. Special cases of a homogeneous integral equation and its solutions are written for different values of the parameter k. Classes are indicated in which the integral equation has a solution. Singular integral equations were considered in works [1–3]. Their kernels were also «incompressiblek, but kernels had an another form. In this connection, the weight classes of the solution existence differ from the class of the solution existence for the equation considered in this work.
Homogeneous differential equation
Integral transform
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Improper integral
Independent equation
Stratonovich integral
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Any equation in which the unknown function appears under the integral sign is called an integral equation. Nonlinear differential equations can also be transformed into integral equations. In fact this is one method used to establish properties of the equation and to develop approximate and numerical solutions. If the unknown function appears in the equation in any way other than to the first power then the integral equation is said to be nonlinear. Only the simplest integral equations can be solved exactly. Usually approximate or numerical methods are employed. The advantage is that integration is a "smoothing operation," whereas differentiation is a "roughening operation". The special convolution equation is a special case of the Volterra equation.
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Volterra equations
Integral transform
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This chapter describes the integral formulation of the equations of motion governing the vibration of continuous systems. An integral equation is an equation in which the unknown function appears under one or more signs of integration. If the unknown function appears nonlinearly in the regular and/or exceptional parts, the equation is said to be a nonlinear integral equation. Based on the type of integral in the regular part, the integral equations are classified as Fredholm- or Volterra-type equations. If the regular part of the integral equation contains a singular integral, the equation is called a singular integral equation. Otherwise, the equation is called a normal integral equation. Several methods, both exact and approximate methods, can be used to find the solutions of integral equations. The chapter considers the method of undetermined coefficients and the Rayleigh-Ritz, Galerkin, collocation, and numerical integration methods.
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