Order Integro-differential Equations Research Articles

EXTENSION OF HAAR WAVELET TECHNIQUES FOR MITTAG-LEFFLER TYPE FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

Fractional order integro-differential equation (FOIDE) of Fredholm type is considered in this paper. The mentioned equations have many applications in mathematical modeling of real world phenomenon like image and signal processing. Keeping the aforementioned importance, we study the considered problem from two different aspects which include the existence theory and computation of numerical approximate solution. FOIDEs have been investigated very well by using Caputo-type derivative for the existence theory and numerical solutions. But the mentioned problems have very rarely considered under the Mittage-Leffler-type derivative. Also, for FOIDE of Fredholm type under Mittage-Leffler-type derivative has not yet treated by using Haar wavelet (HW) method. The aforementioned derivative is non-singular and nonlocal in nature as compared to classical Caputo derivative of fractional order. In many cases, the nonsingular nature is helpful in numerical computation. Therefore, we develop the existence theory for the considered problem by using fixed point theory. Sufficient conditions are established which demonstrate the existence and uniqueness of solution to the proposed problem. Further on utilizing HW method, a numerical scheme is developed to compute the approximate solution. Various numerical examples are given to demonstrate the applicability of our results. Also, comparison between exact and numerical solution for various fractional orders in the considered examples is given. Numerical results are displayed graphically.

Existence and data dependence results for neutral fractional order integro-differential equations

<abstract><p>We assess the multi-derivative nonlinear neutral fractional order integro-differential equations with Atangana-Baleanu fractional derivative of the Riemann-Liouville sense. We discuss results about the existence and difference solution on some data, based on the Prabhakar fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ with generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and the Gronwall-Bellman inequality.</p></abstract>

Second kind Chebyshev collocation technique for Volterra-Fredholm fractional order integro-differential equations

In this work, we present the numerical solution of fractional order Volterra–Fredholm integro-differential equations using the second kind of Chebyshev collocation technique. First, we transformed the problem into a system of linear algebraic equations, which are then solved using matrix inversion to obtain the unknown constants. Furthermore, numerical examples are used to outline the method’s accuracy and efficiency using tables and figures. The results show that the method performed better in terms of improving accuracy and requiring less rigorous work.©2022 JNSMR UIN Walisongo. All rights reserved.

Numerical solution of systems of fractional order integro-differential equations with a Tau method based on monic Laguerre polynomials

In this paper, numerical technique based on monic Laguerre polynomials is proposed to obtain approximate solutions of initial value problems for systems of fractional order integro-differential equations (FIDEs). Operational fractional integral matrix is constructed. This operational matrix is applied together with the monic Laguerre Tau method to solve systems of FIDEs. This systems of FIDEs will be transformed into a system of algebraic equations which can be solved easily. Numerical results and comparisons with other methods are also presented to show the efficiency and applicability of the proposed method.

Numerical Scheme with Convergence Analysis and Error Estimate for Variable Order Weakly Singular Integro-Differential Equation

This paper represents a new application of Legendre wavelet and interpolating scaling function to discuss the approximate solution of variable order integro-differential equation having weakly singular kernel. So far, this technique has been used to solve variable order integro differential equation. In this paper, it is extended to solve variable order integro differential equation with weakly singular kernel. For this purpose, we derive the operational matrices of Legendre wavelets and interpolating scaling function. The resulting operational matrices along with the collocation method transform the original problem into a system of algebraic equation. By solving this system, the approximate solution is obtained. The convergence and error estimate of the presented method have been rigorously investigated. We also discuss the numerical stability of the method. The numerical result of some inclusive examples has been provided through a table and graph for both basis functions that support the robustness and desired precision of the method.

Iterative Method for Solving a Nonlinear Fourth Order Integro-Differential Equation

This study presents the execution of an iterative technique suggested by Temimi and Ansari (TA) method to approximate solutions to a boundary value problem of a 4th-order nonlinear integro-differential equation (4th-ONIDE) of the type Kirchhoff which appears in the study of transverse vibration of hinged shafts. This problem is difficult to solve because there is a non-linear term under the integral sign, however, a number of authors have suggested iterative methods for solving this type of equation. The solution is obtained as a series that merges with the exact solution. Two examples are solved by TA method, the results showed that the proposed technique was effective, accurate, and reliable. Also, for greater reliability, the approximate solutions were compared with the classic Runge-Kutta method (RK4M) where good agreements were observed. For more accuracy the maximum error remainder was found, and the absolute error was computed between the semi-analytical method and the numerical method RK4M. Mathematica® 11 was used as a program for calculations.

A New Numerical Approach Using Chebyshev Third Kind Polynomial for Solving Integrodifferential Equations of Higher Order

There are several classifications of linear Integral Equations. Some of them include; Voltera Integral Equations, Fredholm Linear Integral Equations, Fredholm-Voltera Integrodifferential. In the past, solutions of higher-order Fredholm-Volterra Integrodifferential Equations [FVIE] have been presented. However, this work uses a computational techniques premised on the third kind Chebyshev polynomials method. The performance of the results for distinctive degrees of approximation (M) of the trial solution is cautiously studied and comparisons have been additionally made between the approximate/estimated and exact/definite solution at different intervals of the problems under consideration. Modelled Problems have been provided to illustrate the performance and relevance of the techniques. However, it turned out that as M increases, the outcomes received after every iteration get closer to the exact solution in all of the problems considered. The results of the experiments are therefore visible from the tables of errors and the graphical representation presented in this work.

ALGORITHM FOR THE SOLUTION OF NONLINEAR VARIABLE-ORDER PANTOGRAPH FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS USING HAAR METHOD

This research work is related to establish a powerful algorithm for the computation of numerical solution to nonlinear variable order integro-differential equations (VO-IDEs). The adopted procedure is based on the Haar Wavelet Method (HWM) to compute the required numerical solution to the proposed problem. Further, in the considered problem, a proportional-type delay term is involved, which is also known as the pantograph equation. For a physical problem to investigate the computational purposes, we need to first ensure its existence. For this purpose, we utilize classical fixed results given by Banach and Schauder to establish the sufficient conditions for existence of at least one approximate solution to the proposed problem. Two pertinent examples are given, where the error analysis is also recorded.

Snap-through instability of helicoidal composite imperfect beams surrounded by nonlinear elastic foundation

Bernstein polynomials are exploited, for the first time, in investigation of the nonlinear bending and snap-through instability phenomena of geometrically imperfect bioinspired composite beams with helicoidal orientation scheme. The curved bioinspired beam subjected to different types of lateral loading and embedded on three parameters elastic foundations is considered in formulation and analysis. The nonlinear equilibrium equation of the system based on Euler-Bernoulli beam hypothesis, von Kármán nonlinear strain, and initial imperfect curvature is developed. The equilibrium equation portrayed by nonlinear fourth order integro-differential equation is solved by Bernstein polynomials method and closed form solution of the load-deflection relationship is obtained. The accuracy of the present model and solution procedure is examined by comparing our results with previous results. The developed model is exploited to present the influence of lamination scheme, amplitude of initial imperfection and elastic foundation constants on the nonlinear bending and snap-through/reverse snap-through behaviors in detail. The numerical results show that the Bernstein polynomials approach can capture successfully the nonlinear equilibrium path and overcoming limit (turning) points, which usually appear in the nonlinear response of curved structures. The developed model can be used in design and control the snap-through response and open opportunities for this type of structures in many real applications.

NUMERICAL SOLUTION OF DISTRIBUTED ORDER INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, a numerical algorithm is presented to obtain approximate solution of distributed order integro-differential equations. The approximate solution is expressed in the form of a polynomial with unknown coefficients and in place of differential and integral operators, we make use of matrices that we deduce from the shifted Legendre polynomials. To compute the numerical values of the polynomial coefficients, we set up a system of equations that tallies with the number of unknowns, we achieve this goal through the Legendre–Gauss quadrature formula and the collocation technique. The theoretical aspects of the error bound are discussed. Illustrative examples are included to demonstrate the validity and applicability of the method.

Numerical solution of fractional-order integro-differential equations using Laguerre wavelet method

Laguerre wavelet based numerical method is used for the solution of fractional order Integro-differential equations. This method is based on Laguerre wavelet basis. Laguerre wavelet method (LWM) is then utilized to reduce the fractional Integro-differential equations into the system of algebraic equations. Illustrative examples are represented with tables and figures to check the efficiency and applicability of the proposed technique.

The Numerical Solution For Solving nth Order Integro-Differential Equations via Boubaker Scaling Functions

In this paper, the continuous Boubaker scaling functions were constructed with the presentation on the interval [0,1], which obtained depending on Boubaker polynomials.
 In this current study the Boubaker scaling polynomial has been applied for solving the nth order integro–differential equations (IDE’s).
 The collocation method with the aid of Boubaker scaling functions together were utilized to transform the higher order integro–differential equations into a problem of linear system algebraic equations.
 Some numerical examples were added to show the simplicity and accuracy of the proposed technique. The results have been compared with the exact solution using Matlab and illustrated by graphs.

Solving Fredholm Second Order Integro-Differential Equation with Logarithmic Kernel Using the Airfoil Collocation Method

In this paper, the Airfoil polynomials for solving the second order integro-differential equation with a singular kernel is considered. The collocation method is developed to obtain an approximate solution of the equation. We present an error analysis and conclude by providing numerical tests to verify our results.

Enhanced moving least square method for the solution of volterra integro-differential equation: an interpolating polynomial

This paper presents an enhanced moving least square method for the solution of volterra integro-differential equation: an interpolating polynomial. It is a numerical scheme that utilizes a modified shape function of the conventional Moving Least Square (MLS) method to solve fourth order Integro-differential equations. Smooth orthogonal polynomials have been constructed and used as the basis functions. A robust and unrestricted trigonometric weight function, along with the basis function, drives the shape function and facilitates the convergence of the scheme. The choice of the support size and some controlling parameters ensures the existence of the moment matrix inverse and the MLS solution. Valid explanation and illustration were made for the existence of the inverse linear operator. To overcome problems of near-singularity, the singular value decomposition rule is used to compute the inverse of the moment matrix. Gauss quadrature rule is used to compute the integral at the initial test points when the exact solution is unknown. Some tested problems were solved to show the applicability of the method. The results obtained compare favourable with the exact solutions. Finally, a highly significant interpolating polynomial is obtained and used to reproduce the solutions over the entire problem domain. The negligible magnitude of the error at each evaluation knot demonstrates the reliability and effectiveness of this scheme.

Numerical comparisons for solving fractional order integro-differential equations with non-local boundary conditions

In this paper, univariate Pade approximation is applied to fractional power sries solutions of fractional integro-differential equations with non-local boundary conditions. As it is seen from comparisons, univariate Pade approximation gives reliable solutions and numerical results.

Estimate and Asymptotic Smallness of Solutions of a Weakly Nonlinear Implicit Volterra Integro-Differential Equation of the First Order on the Semiaxis

Estimate and Asymptotic Smallness of Solutions of a Weakly Nonlinear Implicit Volterra Integro-Differential Equation of the First Order on the Semiaxis

New Sufficient Conditions to Ulam Stabilities for a Class of Higher Order Integro-Differential Equations

In this work, we present sufficient conditions in order to establish different types of Ulam stabilities for a class of higher order integro-differential equations. In particular, we consider a new kind of stability, the σ-semi-Hyers-Ulam stability, which is in some sense between the Hyers–Ulam and the Hyers–Ulam–Rassias stabilities. These new sufficient conditions result from the application of the Banach Fixed Point Theorem, and by applying a specific generalization of the Bielecki metric.

Application of New Iterative Method to Fractional Order Integro-Differential Equations

Application of New Iterative Method to Fractional Order Integro-Differential Equations

Effective numerical technique for solving variable order integro-differential equations

Effective numerical technique for solving variable order integro-differential equations

FRACTIONAL POWER SERIES APPROACH FOR THE SOLUTION OF FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

Fractional differential and integral equations are focus of the researchers owing to their tremendous applications in different field of science and technology, such as physics, chemistry, mathematical biology, dynamical system and engineering. In this work, a power series approach called Residual Power Series Method (RPSM) is applied for the solution of fractional (non-integer) order integro-differential equations (FIDEs). The Caputo sense is used for calculating fractional derivatives. Comparison of the obtained solution is made with the Trigonometric Transform Method (TTM) and Optimal Homotopy Asymptotic Method (OHAM). There is no restrictive condition on the proposed solution. The presented technique is simple in applicability and easily computable.