Nonlinear Volterra Integro-differential Equations Research Articles

Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Finite Difference with Quintic B-Splines for Solving a system of nonlinear Volterra Integro-Differential Equations of integer order

This study presents a novel numerical approach for solving systems of nonlinear Volterra Integro-Differential Equations (VIDEs) of integer order by integrating the Finite Difference Method (FDM) with Quintic B-Splines for. The proposed method aims to address the challenges posed by the nonlinearity and integral terms inherent in VIDEs while providing enhanced accuracy and stability in the numerical solution. An error estimate of the approximate solution is proved. Finally, Some example are presented to illustrate the simplicity and the effectiveness of the propose method

Nonlinear Volterra integro-differential equations incorporating a delay term using Picard iterated method

Nonlinear Volterra integro-differential equations incorporating a delay term using Picard iterated method

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel

Numerical solution of fuzzy fractional volterra integro differential equations with boundary conditions

In this study, the boundary value problem of fuzzy fractional nonlinear Volterra integro differential equations of order 1 < ϱ ≤ 2 is addressed. Fuzzy fractional derivatives are defined in the Caputo sense. To show the existence result, the Krasnoselkii theorem from the theory of fixed points is used, where as the well-known contraction mapping concept is utilized in order to show the solution is unique to the proposed problem. Moreover, a novel Adomian decomposition method is utilized to get numerical solution; the approach behind deriving the solution is from Adomian polynomials, and it is organized according to the recursive relation that is obtained. The proposed method significantly decreases the numerical computations by obtaining solutions without the need of discretization or constrictive assumptions. According to the results, there is substantial agreement between the series solutions produced by the fuzzy Adomian decomposition method. Finally, using MATLAB, the symmetry between the lower and upper-cut representations of the fuzzy solutions is demonstrated in the numerical result.

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

Analytical and numerical techniques for solving a fractional integro-differential equation in complex space

&lt;p&gt;In this article, we describe the existence and uniqueness of a solution to the nonlinear fractional Volterra integro differential equation in complex space using the fixed-point theory. We also examine the remarkably effective Euler wavelet method, which converts the model to a matrix structure that lines up with a system of algebraic linear equations; this method then provides approximate solutions for the given problem. The proposed technique demonstrates superior accuracy in numerical solutions when compared to the Euler wavelet method. Although we provide two cases of computational methods using MATLAB R2022b, which could be the final step in confirming the theoretical investigation.&lt;/p&gt;

On stability analysis of a fractional volterra integro delay differential equation in the context of Mittag–Leffler kernel

This paper deals with a nonlinear multi-derivative fractional Volterra integro-differential equation with finite delay involving a non-singular Mittag–Leffler kernel. Krasnoselskii's fixed-point theorem is used to obtain the current result. Also, we establish the generalization of Pachapate's inequality and provide its applications for the existence, uniqueness, stability, and data dependence of the suggested problem. All the results are justified by an example.

Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales

The paper is devoted to studying the existence, uniqueness and certain growth rates of solutions with certain implicit Volterra-type integrodifferential equations on unbounded from above time scales. We consider the case where the integrand is estimated by the Lipschitz type function with respect to the unknown variable. Lipschitz coefficient is an unbounded rd-function and the Banach fixed-point theorem at a functional space endowed with a suitable Bielecki-type norm.

An Iterative Scheme for Solving Arbitrary-Order Nonlinear Volterra Integro-Differential Equations Involving Delay

This paper introduces an iterative-based numerical scheme for solving nonlinear fractional-order Volterra integro-differential equations involving delay. Additionally, we provide sufficient conditions for the existence and uniqueness of the solution. The composite trapezoidal rule is applied to approximate the integral involved in the equation, followed by discretizing the Caputo fractional derivative operator of arbitrary order $$\alpha \in (0,1)$$ by using the classical L1 scheme. Further, the Daftardar-Gejji and Jafari method is employed to solve the implicit algebraic equation. The convergence analysis and error bounds of the proposed scheme are presented. It is shown that the approximate solution converges to the exact solution with order $$( 2-\alpha ).$$ We illustrate the efficacy and applicability of the proposed method through a couple of examples.

Superconvergence and postprocessing of the continuous Galerkin method for nonlinear Volterra integro-differential equations

We propose a novel postprocessing technique for improving the global accuracy of the continuous Galerkin (CG) method for nonlinear Volterra integro-differential equations. The key idea behind the postprocessing technique is to add a higher order Lobatto polynomial of degree k + 1 to the CG approximation of degree k. We first show that the CG method superconverges at the nodal points of the time partition. We further prove that the postprocessed CG approximation converges one order faster than the unprocessed CG approximation in the L2-, H1- and L∞-norms. As a by-product of the postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact. Numerical examples are presented to highlight the superconvergence properties of the postprocessed CG approximations and the robustness of the a posteriori error estimators.

Superconvergent postprocessing of the discontinuous Galerkin time stepping method for nonlinear Volterra integro-differential equations

We propose a very simple but efficient postprocessing technique for improving the global accuracy of the discontinuous Galerkin (DG) time stepping method for solving nonlinear Volterra integro-differential equations. The key idea of the postprocessing technique is to add a higher order generalized Jacobi polynomial of degree k+1 with parameters (−1,0) to the DG approximation of degree k. We prove that the postprocessed DG approximations converge one order faster than the unprocessed DG approximations in the L2-, H1- and L∞-norms. Based on these postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact as the step-size decreases. Numerical experiments are carried out to verify the theoretical results.

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.

ОБ ОЦЕНКЕ СНИЗУ РЕШЕНИЙ СЛАБО НЕЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА С ЗАПАЗДЫВАНИЯМИ

Sufficient conditions are established for lower bounds and tending to infinity as the argument of solutions to a weakly non-linear second-order Volterra integro-differential equation with delays on the half-axis grows unbounded. For this purpose, a non-standard method of reduction to a system, the method of transforming the equations of V. Volterra, the method of partial cutting, the method of integral inequalities with delays and the method of estimating solutions from below Yu.A. Ved are developed. It is shown that solutions with initial data from a certain manifold have lower bounds, and such solutions tend to infinity according to the exponential law with unlimited growth of the argument. This article proposes a technique for solving the problem of estimating from below on the half-line solutions of a weakly nonlinear Volterra-type integro-differential equation of the second order with delays, while the main role is played by the non-standard method of reduction to the system and the method of estimating solutions from below Yu.A.Ved (1965).

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in $$L^2$$ -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm.

Solutions of nonlinear fractional integro - differential equations by combined sumudu transform with adomian decomposition approach

The Caputo formulation is used to handle fractional derivatives in this study. The well-known method form employs Adomian’s polynomials as the proper mathematical tool for dealing with nonlinear variables. The integral equation may be solved by applying the Sumudu transformation to both sides and then the terms can be written out. The solution is given as a convergent infinite series. Nonlinear Volterra integro-differential equations with fractional derivatives are amenable to the proposed method. The results strongly support the validity, applicability, and efficacy of the approach.

Analytical and numerical study of a nonlinear Volterra integro-differential equation with the Caputo-Fabrizio fractional derivative

In this paper we interest concerns an integro-differential equation of Volterra when the derivative of the solution appears under the sign of integration in a non-linear way. The derivation is the Caputo-Fabrizio Fractional Derivative (CF-FD), which increases the interest of this study. we built Lipchitz conditions to obtain the existence and uniqueness of the solution in a Banach space specific to the derivative of Caputo. The analytical study is followed by a complete numerical study.