Order Nonlinear Integro-differential Equations Research Articles

Dynamics of chains of a large number of fully coupled oscillators, as well as oscillators with one-way and two-way delay couplings

This articles considers chains consisting of identical nonlinear equations of second order with couplings in linear elements. It is assumed that there is a large delay in the coupling elements. Moreover, we assume that the chain possesses a large number of elements. Therefore, instead of the original system with many elements, we can study a second order nonlinear integro-differential equation with periodic boundary conditions. The main study is focused on the local dynamics of chains with one-sided, two-sided couplings, as well as fully coupled chains. Also, we study the stability of equilibrium states and identify the critical cases. The condition of a sufficiently large delay helps to determine the parameters for the implementation of critical cases explicitly. Our methodology is based on the infinite-dimensional normalisation method proposed by the author, namely the method of quasinormal forms. We construct quasinormal forms for the chains under consideration that determine the asymptotics of the leading terms of the asymptotic expansions of the solutions to the original system.

A Qualitative Study on Fractional Logistic Integro-differential Equations in an Arbitrary Time Scale

This manuscript deals with the investigation related to uniqueness and existence of solution of fractional order nonlinear pantograph integro-differential equation in arbitrary time scale. The fractional derivatives are defined in Riemann-Liouville sense, the primary tools are taken as Banach contraction principle and Schauder’s fixed point theory to establish the theoretical outcomes. Finally, we give examples to show the efficiency of our results.

On existence and numerical solution of higher order nonlinear integro-differential equations involving variable coefficients

We investigate the approximate solution to a class of fourth-order Volterra–Fredholm integro differential equations (VFIDEs). Additionally, we are able to get some adequate results for the existence of a solution with the use of nonlinear analysis techniques. The basis for the required numerical computation is provided by the Haar wavelet collocations (HWCs) technique, which converts the problems into a system of algebraic equations. Then, in order to obtain the required numerical solution, we solve the resulting system of algebraic equations using Gauss elimination and Broyden’s techniques. This method can be used to show convergence, and the predicted rate of convergence is two. Along with relevant examples, we have included a graphical presentation to demonstrate our suggested approach.

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

A Quantitative Approach to $$n{\text {th}}$$-Order Nonlinear Fuzzy Integro-Differential Equation

A Quantitative Approach to $$n{\text {th}}$$-Order Nonlinear Fuzzy Integro-Differential Equation

EULER AND TAYLOR POLYNOMIALS METHOD FOR SOLVING VOLTERRA TYPE INTEGRO DIFFERENTIAL EQUATIONS WITH NONLINEAR TERMS

In this study, the first order nonlinear Volterra type integro-differential equations are used in order to identify approximate solutions concerning Euler polynomials of a matrix method based on collocation points. This method converts the mentioned nonlinear integro-differential equation into the matrix equation with the utilization of Euler polynomials along with collocation points. The matrix equation is a system of nonlinear algebraic equations with the unknown Euler coefficients. Additionally, this approach provides analytic solutions, if the exact solutions are polynomials. Furthermore, some illustrative examples are presented with the aid of an error estimation by using the Mean-Value Theorem and residual functions. The obtained results show that the developed method is efficient and simple enough to be applied. And also, convergence of the solutions of the problems were examined. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB.

Numerical solution of the second order linear and nonlinear integro-differential equations using Haar wavelet method

In this paper, a numerical algorithm is developed for the solution of second order linear and nonlinear integro-differential equations. The Haar collocation technique is applied to second order linear and nonlinear integro-differential equations. In Haar technique, the second order derivative in both linear and nonlinear integro-differential equation is approximated using Haar functions and the process of integration is used to obtain the expression of first order derivative and expression for the unknown function. Some linear and nonlinear examples are taken from literature for checking validation and convergence of proposed technique. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and gauss points. The convergence rate using different numbers of collocation points is also calculated, which is approximately equal to 2.

On positive solutions of higher order nonlinear fractional integro-differential equations with boundary conditions

On positive solutions of higher order nonlinear fractional integro-differential equations with boundary conditions

Numerical solution of the static beam problem by Bernoulli collocation method

We propose a numerical scheme to obtain an approximate solution of a boundary value problem for fourth order nonlinear integro-differential equation of Kirchhoff type. We first reduce the problem to a nonlinear finite dimensional system based on the Bernoulli polynomial approximation and then solve it by an iterative process together with a collocation method. Convergence of the iterative process and error estimates of the approximate solution are provided. Numerical experiments are conducted to illustrate the performance of the proposed method.

On solvability of a class of nonlinear second order integro-differential equations in the space W 1 2 (ℝ+)

In this paper we study an initial-boundary value problem for a second order nonlinear integro-differential equation with Hammerstein type noncompact operator on the positive semi-axis. This problem has a direct application in the nonlocal problems of physical kinetics. We prove the existence of a nonnegative (nontrivial) solution of this equation in the space W12 (ℝ+). Some special examples of these equations are also discussed, which represent independent theoretical and applied interest.

On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

This paper is devoted to the study of the well-posedness of an initial boundary value problem for an odd higher order nonlinear pseudohyperbolic integrodifferential partial differential equation. We associate to the equationnnonlocal conditions andn+1classical conditions. Upon some a priori estimates and density arguments, we first establish the existence and uniqueness of the strongly generalized solution in a class of a certain type of Sobolev spaces for the associated linear mixed problem. On the basis of the obtained results for the linear problem, we apply an iterative process in order to establish the well-posedness of the nonlinear problem.

Variational iteration algorithm-II for solving the system of third order non-linear integro-differential equations

Variational iteration method is widely used for solving nonlinear, integro-differential equations. Through careful investigation of the iteration formulas of the earlier variational iteration algorithm (VIM), we find unnecessary repeated calculations in each iteration. To overcome this shortcoming, the variational iteration algorithm-II (He et al., 2010) will be used in this paper to solve the system of non linear integro-differential equations. Beside this, the comparison of the exact solution with approximated solution by VIM-II is illustrated by the graphs. Several examples are given to verify the reliability and efficiency of the method. Key words: Variational iteration algorithm-II (VIM), system of nonlinear integro-differential equations.

Second order Volterra-Fredholm functional integrodifferential equations

This paper deals with the study of global existence of solutions to initial value problem for second order nonlinear mixed Volterra-Fredholm functional integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solution .

On the strong maximum principle for second order nonlinear parabolic integro-differential equations

This paper is concerned with the study of the Strong Maximum Principle for semicontinuous viscosity solutions of fully nonlinear, second-order parabolic integro-differential equations. We study separately the propagation of maxima in the horizontal component of the domain and the local vertical propagation in simply connected sets of the domain. We give two types of results for horizontal propagation of maxima: one is the natural extension of the classical results of local propagation of maxima and the other comes from the structure of the nonlocal operator. As an application, we use the Strong Maximum Principle to prove a Strong Comparison Result of viscosity sub- and supersolution for integro-differential equations.

Integral boundary value problems for first order nonlinear impulsive functional integro-differential differential equations

In this paper, we consider the existence of solutions for first order nonlinear impulsive functional integro-differential equations with integral boundary value conditions. We firstly build a new comparison theorem. Then by utilizing the monotone iterative technique and the method of lower and upper solutions, we obtain the existence of extremal solutions or quasi-solutions.

Local and global existence of solutions for a class of integro-differential equations with delay

In this article we consider a class of first order nonlinear integro-differential equations with delay. We propose new approach for investigating local and global existence of the solutions of its Cauchy problem. This approach gives new results.

Existence results for fractional impulsive integrodifferential equations in Banach spaces

This paper is mainly concerned with the existence of solutions of first order nonlinear impulsive fractional integrodifferential equations in Banach spaces. The results are obtained by using fixed point principles. Further, some interesting examples are presented to illustrate the theory.

Global existence of mild solutions of second order nonlinear Volterra integrodifferential equations in Banach spaces

In this paper, we study the existence and uniqueness of mild solutions for second order initial value problems, with nonlocal conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family.

Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type

We discuss the existence of minimal and maximal solutions for a class of first order nonlinear impulsive functional integro-differential equations of mixed type with anti-periodic boundary conditions. The main tool of study is the monotone iterative technique.