Linear Volterra Integral Equations Research Articles

Meshless Technique Based on Moving Least Squares Approximation for Numerical Solutions of Linear and Nonlinear Third-Kind VIEs

The principal objective of this paper is to propose a numerical scheme to solve linear and nonlinear third-kind Volterra integral equations (VIEs). The method approximates the solution using the collocation method based on moving least squares (MLS) approximation. The suggested scheme is meshless, since it needs no background mesh or cell structures, and is thus independent of the geometry of the domain. The approach reduces the solution of third-kind VIEs to the solution of systems of algebraic equations. All integrals appearing in the approach are calculated approximately by the Gauss–Legendre integration formula. The convergence analysis of the proposed scheme has been conducted, and its performance has been tested through a series of numerical examples, demonstrating its effectiveness and applicability in comparison to the existing methods.

Solving Linear Second Kind Non-Homogenous Volterra Integral Equations By Numerical Methods VIM, ADM And TSM

Solving Linear Second Kind Non-Homogenous Volterra Integral Equations By Numerical Methods VIM, ADM And TSM

Numerical Solution of Linear Second-Kind Convolution Volterra Integral Equations Using the First-Order Recursive Filters Method

A new numerical method for solving Volterra linear convolution integral equations (CVIEs) of the second kind is presented in this work. This new approach uses first-order infinite impulse response digital filters method (IIRFM). Three convolutive kernels were analyzed, the unit kernel and two singular kernels: the logarithmic and generalized Abel kernels. The IIRFM is based on the combined use of the Laplace transformation, a first-order decomposition, and a bilinear transformation. This approach often leads to simple analytical expressions of the approximate solutions, enabling efficient numerical calculation, even using single-precision floating-point numbers. When compared with the method of homotopic perturbations with Laplace transformation (HPM-L), the IIRFM approach does not present, in linear cases, the convergence difficulties inherent to iterative approaches. Unlike most solution methods based on the Laplace transform, the IIRFM has the dual advantage of not requiring the calculation of the Laplace transform of the source function, and of not requiring the systematic calculation of inverse Laplace transforms.

On Some Linear Two-Dimensional Volterra Integral Equations of the First Kind

On Some Linear Two-Dimensional Volterra Integral Equations of the First Kind

A multiparametric family of solutions to the Volterra linear integral equation of the first kind

We study the Volterra integral equation of the first kind with an integral operator of order n, a singularity and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms on the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term, we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows one to construct a particular and general solutions to the integro-differential equation in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multi-parameter family of solutions is being constructed for the original integral equation.

Regularization of Solutions of One Class of Systems of Linear Volterra Integral Equations of the First Kind in the Space of Differentiable Vector Functions on the Axis

Regularizing operators according to Lavrentiev are constructed and sufficient conditions for the uniqueness of solutions to systems of linear Volterra integral equations of the first kind on the axis are established.

Numerical methods for highly oscillatory Volterra integral equations with general oscillators

In this paper, collocation methods for linear Volterra integral equations with highly oscillatory trigonometric kernels are considered. Different from the cases with linear oscillator, we need a further action to obtain a practical scheme for general oscillators. To deal with it, the moment free Filon type method is employed in the exact collocation equation resulting to a fully discretized one. In order to conduct the convergence analysis, we firstly study the highly oscillatory integrals occurring in the error equations. Both the theoretical and numerical results show that our method not only converges with respect to step length but also has an asymptotic order which could be two. The asymptotic order indicates that the numerical results will become more accurate as the increase of frequency.

Stability of solutions of systems of Volterra integral equations

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed.

Numerical Solution of Linear Volterra Integral Equation of the Second Kind with Delay Using Lagrange Polynomials

In this study, the linear Volterra integral problem of the second kind will be treated with delay using a Lagrange polynomial. The Volterra integral problem is solved numerically using the chosen technique to obtain the best approximation. Additionally, the test examples are provided to demonstrate, through comparison with other methods' outcomes, the great degree of accuracy of the approximative solutions. Moreover, To verify the accuracy of the calculations that is used in these test examples, the absolute error is used to compare it to the exact solution. For this method, the program is written by MATLAB R2018a language.

A Numerical Method for the Solution of Linear Volterra Integral Equations

The goal of this paper is to present a numerical method for solving the Volterra integral equations using the collocation approximation method with a power series polynomial. The modeled problem is converted into an algebraic equation system and solved using the standard collocation points. After establishing the approach's uniqueness and convergence, numerical examples were used to evaluate its effectiveness. The results show that the method outperforms others. Approximate solution, Integral equations, Polynomial power series, Volterra

Numerical solutions of a class of linear and nonlinear Volterra integral equations of the third kind using collocation method based on radial basis functions

Numerical solutions of a class of linear and nonlinear Volterra integral equations of the third kind using collocation method based on radial basis functions

Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

Initial–boundary value problems for a linear diffusion–advection–reaction equation are considered, with general nonhomogeneous linear boundary conditions and general linear nonlocal boundary conditions. Analytical solutions are obtained using an embedding method. The solutions are expressed in terms of time‐varying functions that satisfy coupled linear Volterra integral equations of the first kind. A boundary element method is applied to numerically solve the integral equations. Three examples are given to demonstrate the accuracy of the numerical solutions when compared with the analytical solutions. The embedding method is applicable to problems with bounded and unbounded spatial domains.

Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

This article proposes a new approach for solving linear Volterra integral equations with variable coefficients using the Reproducing Kernel Method (RKM). This method eliminates the need for the Gram-Schmidt process. However, the accuracy of RKM is influenced by various factors, including the selection of points, bases, space, and implementation method. The main objective of this article is to introduce a generalized method based on the Reproducing Kernel, which is successful in solving a special type of singular weakly nonlinear boundary value problems (BVPs). The easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy of the present method are interesting. The conformity of numerical results, including tables and figures, with theorems related to error analysis and convergence order, confirms the practicality of the present method.

Паралельний алгоритм розв’язування систем інтегральних рівнянь Вольтерри II роду

The problem of increasing the effectiveness of the study of integral models of dynamic systems is considered. A parallel algorithm for solving a system of Volterra linear integral equations of the second kind based on the quadrature method of numerical integration is proposed. The algo-rithm was implemented in the MATLAB computer mathematics system in the form of an m-function. The program uses the MATLAB Distributed Computing Toolbox infrastructure to manage workflows and distribute computations between them on multi-core processors. Com-putational experiments were conducted on a model example using the Symbolic Math Toolbox package for symbolic calculations and a comparison of the execution of parallel calculations with the execution time of the implementation of a sequential algorithm. The results showed a significant increase in the speed of research of integral models on multi-core processors when using the proposed algorithm and its computer implementation.

Singular asymptotic expansion and Legendre collocation method for two-term weakly singular Volterra integral equation of the second kind

This paper develops a singularity-separation Legendre collocation method for solving two-term linear Volterra integral equation with algebraic and logarithmic singular kernels. The psi-series solution at the origin is obtained by an efficient Picard iteration, which is very accurate near the origin. By splitting the interval into a singular subinterval and a regular one, the equation is converted to one with a sufficiently smooth solution on the regular interval, which is discretized by the Legendre collocation method. The efficient computational process is discussed in detail, and the convergence under the square norm is proved. Finally, three numerical examples, including a Caputo fractional differential equation and a Basset equation with initial conditions, are provided to illustrate the efficiency and accuracy of the proposed singularity-separation method.

Автоматизовані засоби тестування програмних модулів розв’язування інтегральних рівнянь Вольтерри другого роду

The article is part of the methodology of automated testing of software modules for solving Volterra integral levels of the second kind. For the implementation of automated testing, the Matlab software environment was selected, which has a wide range of software testing capabilities, in particular: functions for generating data sets for testing; functions for comparing test results; functions for generating test reports, etc. For the development of automated testing tools, the Unit Testing Framework was selected, which is a component of the MATLAB Test Framework, has many ready-made methods for checking the correctness of values and the formation of statistical errors. A set of test problems has been developed for Volterra integral equations of the second kind, which are divided into different types, including linear Volterra integral equations of the second kind, which consist of a kernel with power, exponential, hyperbolic, logarithmic, trigonometric, inverse trigonometric functions and their combination. Developed testing tools used for automated quality control of software modules built on the basis of left rectangle, right rectangle, trapezoidal, and Simpson methods. The developed set of test tasks covers a wide range of possible operating conditions of software modules. The results of the testing allowed to improve the existing software modules to achieve the set conditions for their operation. The conducted research should contribute to the development of more reliable and efficient software modules for solving Volterra integral levels of the second kind. The obtained results are the basis for further research, which will be used in the following directions: development of testing methods for more complex types of Volterra integral levels of the second kind, including the level with nonlinear and non-stationary kernels.

Approximate Analytical Solution of Fuzzy Linear Volterra Integral Equation via Elzaki ADM

In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy solutions are determined using Elzaki ADM after the fuzzy linear Volterra integral equations are first translated into two crisp integral equations utilizing the fuzzy number in parametric form. Three instances of the considered equations are solved to show the established scheme’s dependability, efficacy, and application. The results have a substantial impact on the fuzzy analytical dynamic equation theory. The comparison of the data in a graphical and tabular format demonstrates the robustness of the defined regime. The lower and upper bound solutions’ theoretical convergence and error estimates are highlighted in this paper. A tolerable order of absolute error is also obtained for this inquiry, and the consistency of the outcomes that are approximated and accurate is examined. The regime generated effective and reliable results. The current regime effectively lowers the computational cost, and a faster convergence of the series solution to the exact answer is signaled.

Numerical solution of stochastic Volterra integral equations based on uniform Haar wavelets by using direct method

Nowadays, wavelets have played a crucial role in approximating the solution of a wide range of problems arising in science and engineering. Wavelets have been used in numerous areas of applied mathematics as diverse as signal analysis, statistics, computer-aided geometric, image processing and numerical analysis. The uniform Haar wavelets are most widely used wavelets for solving stochastic integral equations, outperforming Legendre, Daubechies, etc. The uniform Haar wavelets are generated from pairs of piecewise constant functions and can be simply integrated. Furthermore, the uniform Haar functions are orthogonal and form a good transform basis. In this paper, we have proposed an efficient numerical method to solve a class of stochastic linear Volterra integral equations (SLVIEs) based on the uniform Haar wavelets. The aim of this paper is to apply numerical method to obtain approximate solution of SLVIEs. By using this method, the SLVIEs can be reduced to a linear system of algebraic equations. Also, the error analysis of this method is performed. Finally, some numerical examples are given to demonstrate pertinent features of this method. Furthermore, we compare obtained results from this method with the achieved results from relevant studies.

Iterative schemes for linear equations of the second kind and related inverse problems

This paper consists of two parts. The first one deals with the generation of an iterative algorithm to obtain an approximate solution of a linear equation of the second kind in a Banach space. This generation is based on a perturbed version of the geometric series theorem which, in particular, allows us to find a family of unisolvent linear Fredholm integral equations of the second kind, even when the associated linear operator has norm greater than or equal to 1. When we consider Fredholm equations of this type and linear Volterra integral equations of second kind, the numerical schemes obtained when appropriate Schauder bases are also introduced in the spaces where the equations operate, enable us to approximate their respective solutions iteratively. The second part of this work focuses on the design of a numerical method for solving an inverse problem associated with a linear equation of the second kind in a Banach space, a method which we apply to problems of parameter estimation related to the two classes of integral equations mentioned above.

Solving a singular integral equation for the one-dimensional Coulomb problem

A new integral equation that describes the behavior of the momentum space wave function for the one-dimensional Coulomb potential is proposed. The obtained result turned out to be a homogeneous Fredholm integral equation of the second kind and a singular integral equation, because its kernel has a singularity at some point in the momentum space. A nontriviality of the method of solving this singular integral equation lies in the application of the integral representation for its integral kernel. The technique applied in this paper made it possible to show that the wave function in the momentum representation is simultaneously a solution of the homogeneous Fredholm integral equation of the second kind and of the linear Volterra integral equation of the second kind. Since a linear Volterra integral equation of the second kind was easily transformed into a second order linear inhomogeneous differential equation with constant coefficients, the eigenfunctions and eigenvalues in the one-dimensional Coulomb problem were found without any difficulties. Such a circumstance may indicate the validity of the new integral equation and the proposed method of its solving.