Linear Volterra Integro-differential Equations Research Articles

Study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments

In this paper, we give new results on the study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments, considered in a Banach space. Using the method suggested by Hernández et al. (2010), which is based on the theory of resolvent operators for integral equations and fixed point theorems, we prove the main results on the existence and uniqueness of the mild solution of this type of problems which have not been studied before. We also give an example to which our main results are applicable.

Numerical treatment of linear Volterra integro differential equations using variational iteration algorithm with collocation

To numerically solve the linear Volterra integro-differential equation, this study employs fourth-kind Chebyshev polynomials and the variational iteration algorithm with collocation, which is a combination of the variational iteration strategy and collocation technique. By applying fourth-kind Chebyshev polynomials to the variational iteration method with collocation for solving Volterra integro-differential equations, mathematical problems with a broad range of multidisciplinary applications are addressed, and numerical techniques that produce more accurate and efficient results are developed. The recommended method is then used, and the fourth-kind Chebyshev polynomials generated for the given integro-differential equation serve as the trial functions for the approximation. As a result, the suggested method's significance probably goes beyond a particular equation or application, as it contributes to the larger field of mathematical modeling and numerical analysis. Research methods employing a variational iteration algorithm with collocation aim to provide general techniques that can be applied to a wide range of problems. Additionally, numerical examples were provided to highlight the applicability and dependability of the proposed methodology. The mathematical computations were carried out using the Maple 18 software.

ОБ АСИМПТОТИЧЕСКОЙ УСТОЙЧИВОСТИ РЕШЕНИЙ ЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА В СЛУЧАЕ НЕДИФФЕРЕНЦИРУЕМОСТИ СРЕЗАННЫХ ФУНКЦИЙ

Sufficient conditions are established for the asymptotic stability of any solution of a linear second-order Volterra integro-differential equation in the case when the truncated functions may be non-differentiable at some points of the semiaxis. To do this, first, using a non-standard method of reduction to a system, the equation under consideration is reduced to an equivalent system, and the method of transforming equations and the method of cutting functions are applied to this system. The truncated kernel and free term are associated with the coefficient of the new unknown function using the inequality between the geometric and arithmetic means of two functions. Using the method of integral inequalities, sufficient conditions are established for the boundedness on the semiaxis of all solutions of the system. Then the method of squaring equations is used to the first equation of the system and the Lyusternik-Sobolev lemma is applied. Finally, based on a non-standard substitution, we complete the proof of the asymptotic stability of solutions to a given second-order linear integro-differential equation. An illustrative example is given.

Solution space characterisation of perturbed linear Volterra integrodifferential convolution equations: The [formula omitted] case

In this paper we characterise the Lp stability of the following perturbed linear Volterra integrodifferential convolution equation, x′(t)=∫[0,t]ν(ds)x(t−s)+f(t),t≥0,where ν is a finite signed Borel measure. Additionally we provide a framework which points to necessary and sufficient conditions on the forcing function that ensures the solution lies in a particular function space. We highlight how such a result is of interest when studying perturbed Stochastic Functional Differential Equations (SFDEs).

Approximate Solution of Linear Volterra Integro-Differential Equation by Touchard Polynomials Method

The idea of this research is to find approximate solution of Linear Volterra Integro-Differential Equation (LVIDE) of the first order and second kind by using different degrees of the Touchard polynomials is presented. Because the simplicity definition applying this method will give high resolution results in low cost and short time, also can be easily differentiated, integrated and converge to an exact solution.The algorithm and example given are to illustrate the solution in this way and compare it with the exact solution.

Nonlinear viscoelastic damping in a Poynting oscillator

This work considers a cylindrical rod, acting as a nonlinear viscoelastic spring, with an attached mass that can displace along and rotate about the rod’s axis, thereby inducing an extension and twist in the rod. The coupling between them produces the Poynting effect in the body. The material is modeled by the nonlinear single integral Pipkin–Rogers constitutive equation. The mass has been assumed to be at rest in a long-time equilibrium state and then given a small axial or rotational disturbance and released (plucked). The subsequent motion is governed by a linear Volterra integro-differential equation. For a particular choice of material parameters, analytical expressions for the time-dependent decay of the disturbance are obtained in terms of the extension/torsion coupling, material stiffness, the amount and rate of stress relaxation, and the inertia of the mass. For other choices of material parameters, the results provide insight into how these quantities affect the time-dependent decay. A number of plucking scenarios are treated such as extensional or rotational disturbances from the undeformed state, rotational plucking after a finite axial stretch, and extensional plucking after a finite rotation. Simultaneous rotational and extensional plucking leads to a system of Volterra integro-differential equations whose treatment is deferred to later work.

On the Method of Lyapunov Functionals for a Linear First-Order Volterra Integrodifferential Equation with Delay on the Semiaxis

On the Method of Lyapunov Functionals for a Linear First-Order Volterra Integrodifferential Equation with Delay on the Semiaxis

SEE Transform Technique for Solving System of Linear Volterra Integro-Differential Equations of the Second Kind

There are many integral transforms that are widely used to solve numerous real-life, science, and engineering problems. In this article, we present SEE transform for determining the solution of the system of linear Volterra integro-differential equations of the second kind. Some applications have been given and solved by using the SEE transform for illustrating the applicability of the SEE transform. Results of the applications assert that the SEE transform is very effective for obtaining the exact solution of this equation.

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Abstract In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.

Fractional Linear Volterra Integro-Differential Equations in Banach Spaces

The paper presents the foundations of the theory of linear fractional Volterra integro-differential equations of convolution type in Banach spaces. It is established that the existence of a fractional resolvent operator for such equations is equivalent to the well-posedness of the formulation of the initial problem for them. Within the framework of this approach, a theorem of the Hille–Yosida type is proved.

Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order [Formula: see text] in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels

On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels

Taylor collocation method for high-order neutral delay Volterra integro-differential equations

In this paper, the Taylor collocation method is applied to numerically solve a kth-order neutral linear Volterra integro-differential equation with constant delay and variable coefficients.We also provide a rigorous analysis to estimate the difference between the exact and approximate solution and their derivatives up to order k-1. Numerical examples are included to prove the performance of the presented convergent algorithm.

Approximation Technique for Solving Linear Volterra Integro-Differential Equations with Boundary Conditions

This paper presents a new technique for solving linear Volterra integro-differential equations with boundary conditions. The method is based on the blending of the Chebyshev spectral methods. The application of the proposed method leads the Volterra integro-differential equation to a system of algebraic equations that are easy to solve. Some examples are introduced and the obtained results are compared with exact solution as well as the methods that reported in the literature to illustrate the effectiveness and accuracy of the method. The results demonstrate that there is congruence between the numerical and the exact results to a high order of accuracy. Tables were generated to verify the accuracy convergence of the method and error. Figures are presented to show the excellent agreement between the results of this study and the results from literature.

Solving the second kind linear volterra integrodifferential equations using the complex SEE integral transform

As an important type of integral equation, Volterra integral equations snatch the focus of many scientists and mathematicians to provide approximate or exact solutions to such equations. The integral transform capability of providing an algebraic solution to the integral equations led the mathematical community to lean heavily on them to solve those kinds of equations, including the Volterra integrodifferential equations of the second kind. This paper uses the complex SEE integral transform to find the exact solution to the second kind linear Volterra integrodifferential equation. The capability and efficiency of complex SEE integral transform in providing an exact solution with the minimum number of computations possible are demonstrated via practical applications.

Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

In this article, an effective approach has been proposed to obtain the approximate solutions of linear and nonlinear two-dimensional Volterra integro-differential equations. First, the two-dimensional wavelets are introduced, and using it the operational matrices of integration, differentiation, and product have been constructed. Then, by utilizing the properties and matrices of wavelets along with the collocation point, the matrix form of Volterra integro-differential equations has been derived. This approach reduces the two-dimensional linear and nonlinear Volterra integro-differential equations into the system of linear and nonlinear algebraic equations respectively. The convergence analysis and error analysis have been extensively studied by the help of two-dimensional wavelets approximation. Some illustrative examples are examined to clarify the accuracy and effectiveness of the proposed scheme. The graphical representations obtained by two proposed wavelets have been plotted to justify the applicability and validity of the method.

Block boundary value methods for solving linear neutral Volterra integro-differential equations with weakly singular kernels

A class of block boundary value methods is constructed for the solution of linear neutral Volterra integro-differential equations with weakly singular kernels. Under suitable conditions on the data, it is shown that these methods yield optimal convergence rates when implemented on special graded meshes. Furthermore, these methods are easily extended to solve linear Volterra integral equations of the 2nd kind with weakly singular kernels. Numerical experiments confirm the theoretical results and the accuracy of the methods, and a comparison with piecewise polynomial collocation methods is provided.

Sumudu Lagrange-spectral methods for solving system of linear and nonlinear Volterra integro-differential equations

This paper presents new efficient numerical methods for solving Volterra integro-differential equations and a system of nonlinear delay integro-differential equations which arises in biology. The principal idea of these approaches is based on a careful blend of the Petrov-Galerkin technique and the Sumudu transform method. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear integro-differential equations, with their associated initial conditions are reduced to linear and nonlinear systems of algebraic equations in the unknown expansion coefficients. Solving the resulting algebraic systems by Gaussian elimination and Newton's methods respectively, approximate solutions of the integro-differential problems are constructed. Detailed error analysis of the proposed methods is carried out to establish and ascertain the reliability and effectiveness of the methods. The methods are then tested on several examples, and the results are compared with those obtained via existing methods in the literature. The numerical results showed that the proposed methods are accurate, efficient, and reliable for solving all kinds of integro-differential equations.

A hybrid method for solution of linear Volterra integro-differential equations (LVIDES) via finite difference and Simpson’s numerical methods (FDSM)

In this paper, a hybrid of Finite difference-Simpson’s approach was applied to solve linear Volterra integro-differential equations. The method works efficiently great by reducing the problem into a system of linear algebraic equations. The numerical results shows the simplicity and effectiveness of the method, error estimation of the method is provided which shows that the method is of second order convergence.

Modified Adomian Decomposition Method for the Solution of Integro-Differential Equations

This paper is concerned with modification of the Adomian Decomposition Method for solving linear and non-linear Volterra and Volterra-Fredholm Integro-Differential equations. The Modified form of ADM was carried out by replacing the Adomian polynomials constructed in the conventional Adomian Decomposition Method with the constructed canonical polynomials. The modified Adomian Decomposition Method was applied to solve some existing example. The results obtained using the newly modified ADM proved superior when compared with the conventional ADM.