Order Integro-differential Equations Research Articles

Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

In this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

Error analysis and approximation of Jacobi pseudospectral method for the integer and fractional order integro-differential equation

Time-space Jacobi pseudospectral method is constructed to approximate the numerical solutions of the fractional Volterra integro-differential and parabolic Volterra integro-differential equations. We define fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi-Gauss-Lobatto points. The fractional derivative is defined in the modified Atangana-Baleanu derivative defined in the Caputo sense formula at JGL points. Further, we transform the domain of fractional Volterra integro-differential and parabolic Volterra integro-differential equations to the standard interval [−1,1] using variable transformation and function transformation. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The theory of error estimates and convergence analysis for the proposed method is also derived. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

Hyers–Ulam stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives

• In this paper, we investigate the boundary value problem with Riemann-Liouville fractional differential equations for a class of impulsive implicit nonlinear fractional order integro-differential equations. • We are using some sufficient conditions to achieve the existence and uniqueness results of the projected model with the help of Banach contraction principle, Schauder’s fixed point theorem and Kranoselskii’s fixed point theorem. • Moreover, we present different types of stability such as Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability using the classical technique of functional analysis. • To illustrate our theoretical results, at the end we give an example. In this article, we investigate the existence, uniqueness, and stability of coupled implicit fractional integro-differential equations with Riemann–Liouville derivatives. We analyze the existence and uniqueness of the projected model with the help of Banach contraction principle, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem. Moreover, we present different types of stability using the classical technique of functional analysis. To illustrate our theoretical results, at the end we give an example.

The radiative transfer model for the greenhouse effect

Radiative transfer is at the heart of the mechanism to explain the greenhouse effect based on the partial infrared opacity of carbon dioxide, methane and other greenhouse gases in the atmosphere. In absence of thermal diffusion, the mathematical model consists of a first order integro-differential equation coupled with an integral equation for the light intensity and the temperature, in the atmosphere. We revisit this much studied system from a mathematical and numerical point of view. Existence and uniqueness and implicit solutions of the Milne problem for grey atmospheres are stated. Numerical simulations are given for grey and non-grey atmospheres and applied to calculate the effect of greenhouse gases. In the context of a transparent atmosphere for sunlight, it is found that by doubling the absorption coefficient in the infrared absorption range of texttt {CO}_2 the temperature decreases by 2%. On the other hand, the same changes but in the low infrared range of the sunlight leads to an increase of temperature in the atmosphere. Several computer codes were written to cross-validate the results. The authors conclude that the radiative transfer model without thermal diffusion for an atmosphere transparent to the incident sunlight is not capable of explaining the greenhouse effect due to the greenhouse gases. A decreasing temperature due to an increasing proportion of texttt {CO}_2 has been observed in the high atmosphere (D.W.J. Thomson et al, nature11579). In the lower atmosphere thermal diffusion and convection cannot be neglected and since the absorption coefficient are highly dependent on the temperature, a full ocean–atmosphere–biosphere climate model is required. Hence, driving conclusions from this study on climate change should be cautiously avoided and a review of the hypothesis of the radiative transfer argument commonly found in textbooks should be revisited.

On a method for solving nonlinear integro differential equation of order

This work is concerned with the study of a general class of nonlinear integro-differential equations of order n. Using a suitable transformation, we derive an equivalent nonlinear Fredholm-Volterra integral equation (NF-VIE) to this class of equations. The existence of continuous solutions for the NF-VIE is investigated subject to the verification of some sufficient conditions. We apply the modified Adomian's decomposition method (MADM) and homotopy analysis method (HAM) to solve this NF-VIE. The convergence and error estimate of the approximate solution are also studied. The numerical results in this article show that the HAM technique may give an approximate solution with high accuracy and convergence rate faster than the one obtained using the MADM technique provided the convergence control parameter \(\hbar\) is properly chosen when applying the HAM.

Solution of third order linear and nonlinear boundary value problems of integro-differential equations using Haar Wavelet method

In this paper, numerical solution of third order integro-differential equation with boundary conditions is given utilizing Haar collocation technique. Both nonlinear and linear integro-differential equations are solved using this method. The third order derivative is approximated using Haar functions in both nonlinear and linear integro-differential equations. Integration is used to obtain the expression of lower order derivatives as well as the solution for the unknown function. The Gauss elimination approach is utilized for linear systems and Broyden approach is adopted for nonlinear systems. Validation and convergence of the proposed approach are illustrated using some examples. At various collocation and gauss points, the maximum absolute and root mean square errors are compared to the exact solution. The convergence rate is also measured using different numbers of nodal points, and it is nearly equal to 2.

Simple approximation for the ruin probability in renewal risk model under interest force via Laguerre series expansion

Although the ruin probability in a renewal insurance risk model with credit interest may be viewed as a classical research problem, exact solutions are only available in the literature in very special cases when both the claim amounts and the interclaim times follow distributions such as the exponential. This is a long standing problem especially from a computational point of view, and the difficulty lies in the fact that the ruin probability usually satisfies a higher order integro-differential equation and/or an ordinary differential equation with non-constant coefficients. In this paper, for a large class of interclaim time distributions (including a combination of exponentials), we shall develop an approximation for the ruin probability using Laguerre series expansion as a function of the initial surplus level. It is shown that the (approximated) Laguerre coefficients can be solved from a system of linear equations, a procedure that is very easy to implement. A main advantage of our approach is that no specific distributional assumption on the claim amounts is required, apart from some mild differentiability and integrability conditions that can be verified. Numerical examples are provided to illustrate the very good performance of our approximation including both light-tailed and heavy-tailed claims.

Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

<p style='text-indent:20px;'>We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – <i>generalized positive definite kernel</i> (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.</p>

A CLASS OF INVERSE PROBLEMS FOR A PARTIAL INTEGRODIFFERENTIAL EQUATIONS OF THE FIRST ORDER

A CLASS OF INVERSE PROBLEMS FOR A PARTIAL INTEGRODIFFERENTIAL EQUATIONS OF THE FIRST ORDER

Existence results of noninstantaneous impulsive fractional integro-differential equation

Abstract Existence of mild solution for noninstantaneous impulsive fractional order integro-differential equations with local and nonlocal conditions in Banach space is established in this paper. Existence results with local and nonlocal conditions are obtained through operator semigroup theory using generalized Banach contraction theorem and Krasnoselskii’s fixed point theorem, respectively. Finally, illustrations are added to validate derived results.

New approach to approximate the solution for the system of fractional order Volterra integro-differential equations

The main aim of this article is the extension of Optimal Homotopy Asymptotic Method to the system of fractional order integro-differential equations. The systems of fractional order Volterra integro-differential equations (SFIDEs) are taken as test examples. The fractional order derivatives are defined in the Caputo fractional form and the optimal values of auxiliary constants are calculated using the well-known method of least squares. The results obtained by proposed scheme are very encouraging and show close resemblance with exact values. Hence it will be more appealing for the researchers to apply the proposed scheme to different fractional order systems arising in different fields of sciences especially in fluid dynamics and bio-engineering.

Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series

In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

EXISTENCE RESULTS AND STABILITY CRITERIA FOR ABC-FUZZY-VOLTERRA INTEGRO-DIFFERENTIAL EQUATION

In the modeling of dynamical problems the fractional order integro-differential equations (IDEs) are very common in science and engineering. The scientists are developing different aspects of these models. The existence of solutions, stability analysis and numerical simulations are the most commonly studied aspects. There is no paper in literature describing the Hyers–Ulam stability (HU-stability) for fuzzy-fractional order models. Therefore, keeping the importance of the study, we consider the existence, uniqueness and HU-stability of a fractional order fuzzy-Volterra IDE.

Implicit Riesz wavelets based-method for solving singular fractional integro-differential equations with applications to hematopoietic stem cell modeling.

Riesz wavelets in have been proven as a useful tool in the context of both pure and numerical analysis in many applications, due to their well prevailing and recognized theory and its natural properties such as sparsity and stability which lead to a well-conditioned scheme. In this paper, an effective and accurate technique based on Riesz wavelets is presented for solving weakly singular type of fractional order integro-differential equations with applications to solve system of fractional order model that describe the dynamics of uninfected, infected and free virus carried out by cytotoxic T lymphocytes (CTL).The Riesz wavelet in this work is constructed via the smoothed pseudo-splines refinable functions. The advantage of using such wavelets, in the context of fractional and integro-differential equations, lies on the simple structure of the reduced systems and in the powerfulness of obtaining approximated solutions for such equations that have weakly singular kernels. The proposed method shows a good performance and high accuracy orders.

English

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On the return to equilibrium problem for axisymmetric floating structures in shallow water

In this paper we address the return to equilibrium problem for an axisymmetric floating structure in shallow water. First we show that the equation for the solid motion can be reduced to a delay differential equation involving an extension–trace operator whose role is to describe the influence of the fluid equations on the solid motion. It turns out that the compatibility conditions on the initial data for the return to equilibrium configuration are not satisfied, so we cannot use the result from Bocchi (2020 SIAM J. Math. Anal. 52 306–39) for the nonlinear problem. Hence, assuming small amplitude waves, we linearize the equations in the exterior domain and we keep the nonlinear equations in the interior domain. For such configurations, the extension–trace operator can be computed explicitly and the delay term in the differential equation can be put in convolution form. The solid motion is therefore governed by a nonlinear second order integro-differential equation, whose linearization is the well-known Cummins equation. We show global in time existence and uniqueness of the solution using the conservation of the total fluid-structure energy.

Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations

This paper is devoted to the numerical scheme for a class of fractional order integrodifferential equations by reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials. Reproducing kernel function in the form of Jacobi polynomials is established for the first time. It is implemented as a reproducing kernel method. The numerical solutions obtained by taking the different values of parameter are compared; Schmidt orthogonalization process is avoided. It is proved that this method is feasible and accurate through some numerical examples.

Boubaker Wavelet Functions for Solving Higher Order Integro-Differential Equations

In the present paper, the properties of Boubaker orthonormal polynomials are used to construct new Boubaker wavelet orthonormal functions which are continuous on the interval [0, 1). Then, a Boubaker wavelet orthonormal operational matrix of the derivative is obtained with the new general procedure. The matrix elements can be expressed in a simple form that reduces the computational complexity. The collocation method of the Boubaker orthonormal wavelet functions together with the application of the derived operational matrix of the derivative are then utilized to transform the higher-order integro-differential equation into a solution of linear algebraic equations. As a result, the solution of the original problem reduces to the solution of a linear system of algebraic equations and can be sufficiently solved by an approximate technique. The main advantage of the suggested method is that the orthonormality property greatly simplifies the original problem and leads to easy calculation of the coefficients of expansion. Special attention is needed to perform the convergence analysis. The error is analyzed when a sufficiently smooth function is expanded in terms of the Boubaker orthonormal wavelet functions, then an estimation of the upper bound of the error is calculated. The results obtained by the technique in the current work are reported by solving some numerical examples and the accuracy is checked by comparing the results with the exact solution.

Numerical Solution of Functional Volterra–Hammerstein Integro-differential Equations of Order Two

In this paper, the functional Volterra–Hammerstein integro-differential equations (FVHIDEs) of order two with variable coefficients are studied. First, the existence and uniqueness of the solution to the mentioned problem are proved. Then a numerical method is proposed to solve the FVHIDEs. The proposed method is a combination of the well-known finite difference method with successive approximations method, trapezoidal quadrature rule and natural cubic spline interpolation method. An error bound is obtained, and the convergence of the method is proved. The numerical stability of the method is also proved by a suitable way. Finally, some examples are given to show the accuracy and efficiency of the proposed method.

Some two-point problems for second order integro-differential equations with argument deviations

In the paper we describe the classes of unique solvability of the Dirichlet and mixed two point boundary value problems for the second order linear integro-differential equation $$ u''(t)=p_0(t)u(t)+p_1(t)u(\tau_1(t))+\int_{a}^{b}p(t,s)u(\tau(s))ds+ q(t). $$% On the basis of the obtained and, in some sense, optimal results for the linear problems, by the a priori boundedness principle we prove the theorems of solvability and unique solvability for the second order nonlinear functional differential equations under the mentioned boundary conditions.