Nonlinear Fractional Integro-differential Equations Research Articles

Infinite delay fractional stochastic integro-differential equations with Poisson jumps of neutral type

This paper addresses the study of nonlinear fractional stochastic neutral integro-differential equations with Poisson jumps and infinite delay in Hilbert space. Initially, the existence and uniqueness of mild solution for the proposed system are verified by applying the suitable conditions of sectorial operators, fixed point technique and stochastic analysis with non-Lipschitz condition and Lipschitz condition as a special case. Also, continuous dependence of mild solutions for proposed model is determined. Finally an example is provided to illustrate the developed theory.

On a Nonlocal Multipoint and Integral Boundary Value Problem of Nonlinear Fractional Integrodifferential Equations

The aim of this paper is to give the existence as well as the uniqueness results for a multipoint nonlocal integral boundary value problem of nonlinear sequential fractional integrodifferential equations. First of all, we give some preliminaries and notations that are necessary for the understanding of the manuscript; second of all, we show the existence and uniqueness of the solution by means of the fixed point theory, namely, Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Last, but not least, we give two examples to illustrate the results.

Analysis of Fractional Integro-Differential Equations with Nonlocal Erdélyi-Kober Type Integral Boundary Conditions

Abstract In this article, we study the existence and uniqueness of solutions for nonlinear fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions. The existence results are based on Krasnoselskii’s and Schaefer’s fixed point theorems, whereas the uniqueness result is based on the contraction mapping principle. Examples illustrating the applicability of our main results are also constructed.

On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses

In this paper, we consider the initial boundary value problem for a class of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses in Banach spaces. Sufficient conditions of existence and uniqueness of PC-mild solutions for the equations are obtained via general Banach contraction mapping principle, Krasnoselskii’s fixed point theorem, and α-order solution operator.

A new numerical scheme for solving pantograph type nonlinear fractional integro-differential equations

In this work, a general class of pantograph type nonlinear fractional integro-differential equations (PT-FIDEs) with non-singular and non-local kernel is considered. A numerical scheme based on the orthogonal basis functions including the shifted Legendre polynomials (SLPs) is proposed. First, we expand the unknown function and its derivatives in terms of the SLPs with unknown coefficients. Then, we present several theorems based on the SLPs for the help to achieve the approximate solution of the problem under study. Finally, by utilizing these theorems together with the collocation points, the main problem is transformed to a system of linear or nonlinear algebraic equations, which can be simply solved. An investigation for error estimate is discussed. The accuracy and efficiency of the proposed scheme are reported by four illustrative examples.

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 theory and semi-analytical study of non-linear Volterra fractional integro-differential equations

In this article, we investigate the semi-analytic solutions of non-linear Volterra fractional integro-differential equations by using Laplace Adomian decomposition method. We discuss the method in general and provide examples for the illustration purpose. Moreover, we give existence and uniqueness result for the solutions using Banach contraction principle as tool. Also we provide some results for the convergence of Laplace Adomian decomposition technique. The results in this work show that the method of Laplace Adomian decomposition is an effective tool for the solutions of Volterra fractional integro-differential equations.

The Asymptotic Behavior of Solutions of a Fractional Integro-differential Equation

In this paper, we study the asymptotic behavior of solutions for an initial value problem with a nonlinearfractional integro-differential equation. Most of the existing results in the literature assume the continuity of theinvolved kernel. We consider here a kernel that is not necessarily continuous, namely, the kernel of the RiemannLiouville fractional integral operator that might be singular. We determine certain sufficient conditions underwhich the solutions, in an appropriate underlying space, behave eventually like power functions. For this purpose,we establish and generalize some well-known integral inequalities with some crucial estimates. Our findings aresupported by examples and numerical calculations.

Numerical solution of multi-variable order fractional integro-differential equations using the Bernstein polynomials

Integro-differential equations are developed as models in enormous fields of engineering and science such as biological models, population growth, aerospace systems and industrial mathematics. In this work, we consider a general class of nonlinear fractional integro-differential equations with variable order derivative. We use the operational matrices based on the Bernstein polynomials to obtain numerical solution of this type of equations. By utilizing the operational matrices along with the Newton–Cotes collocation points, the problem under study is reduced to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, some examples are included to show the accuracy and validity of the proposed method.

Existence of solutions for nonlinear fractional integro-differential equations

In this paper, by means of the Krasnoselskii fixed point theorem, the existence of solutions for a boundary value problem of nonlinear sequential fractional integro-differential equations are investigated. Two examples are given to illustrate our results.

Approximate Analytical Solution for a Coupled System of Fractional Nonlinear Integrodifferential Equations by the RPS Method

In this work, a modified residual power series method is implemented for providing efficient analytical and approximate solutions for a class of coupled system of nonlinear fractional integrodifferential equations. The proposed algorithm is based on the concept of residual error functions and generalized power series formula. The fractional derivative is described under the Caputo concept. To illustrate the potential, accuracy, and efficiency of the proposed method, two numerical applications of the coupled system of nonlinear fractional integrodifferential equations are tested. The numerical results confirm the theoretical predictions and depict that the suggested scheme is highly convenient, is quite effective, and practically simplifies computational time. Consequently, the proposed method is simple, accurate, and convenient in handling different types of fractional models arising in the engineering and physical systems.

The existence and Ulam–Hyers stability results for $$\psi $$-Hilfer fractional integrodifferential equations

We establish sufficient conditions for the existence and uniqueness of solutions for a class of nonlinear fractional integrodifferential equations with boundary conditions involving $$\psi $$ -Hilfer fractional derivative of order $$0<\alpha <1 $$ and type $$0\le \beta \le 1$$ . Different types of Ulam–Hyers stability for solutions of the given problem are also discussed. The desired results are proved in weighted spaces with the aid of fixed point theorems due to Schauder, Schaefer and Banach together with generalized Gronwall inequality. Examples illustrating the obtained results are presented.

Existence and Uniqueness of Mild Solutions for Fractional Partial Integro-Differential Equations

In this paper, we study a class of nonlinear time fractional partial integro-differential equations. The main results for the problem are obtained by the measure of noncompactness, solution operator, convex-power condensing operator and general Banach contraction mapping principle.

A stable least residue method in reproducing kernel space for solving a nonlinear fractional integro-differential equation with a weakly singular kernel

A stable least residue method for solving a nonlinear fractional integro-differential equation with a weakly singular kernel in the reproducing kernel space is proposed. To solve the equation, the multiwavelets bases in the reproducing kernel space are constructed based on the cubic Legendre wavelets in L2[0,1]. The best approximate solution could be obtained by solving the normal equation. Meanwhile, we provide the adaptive convergence order and the stability proof of the scheme. It is encouraging that the new method is stable and the accuracy of the method is preserved even in the case if the solution has fast oscillations. Four examples strongly demonstrate the validity of the scheme.

On Hybrid Type Nonlinear Fractional Integrodifferential Equations

In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as the uniqueness of the corresponding approximate solutions by using hybrid fixed point theorems and provide upper and lower bounds to these solutions. Furthermore, some examples are provided, in which the general claims in the main theorems are demonstrated.

On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray–Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work.

Operational matrices based on hybrid functions for solving general nonlinear two-dimensional fractional integro-differential equations

In the present paper, an attempt was made to develop a numerical method for solving a general form of two-dimensional nonlinear fractional integro-differential equations using operational matrices. Our approach is based on the hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials. Error bound and convergence analysis of the proposed method are discussed. We prove that the order of convergence of our method is $$ O( \frac{1}{2^{2M-1}N^{M}M!} ) $$ . The presented method is tested by seven test problems to demonstrate the accuracy and computational efficiency of the proposed method and to compare our results with other well-known methods. The comparison highlighted that the proposed method exhibits superior performance than the existing methods, even using a few numbers of bases.

Cauchy problem for fractional non-autonomous evolution equations

This paper deals with the following Cauchy problem to nonlinear time fractional non-autonomous integro-differential evolution equation of mixed type via measure of noncompactness $$ \left\{\begin{array}{ll} ^CD^{\alpha}_tu(t)+A(t)u(t)= f(t,u(t),(Tu)(t), (Su)(t)),\quad t\in [0,a], \\[12pt] u(0)=A^{-1}(0)u_0 \end{array} \right. $$ in infinite-dimensional Banach space $E$, where $ ^CD^{\alpha}_t$ is the standard Caputo's fractional time derivative of order $0<\alpha\leq 1$, $A(t)$ is a family of closed linear operators defined on a dense domain $D(A)$ in Banach space $E$ into $E$ such that $D(A)$ is independent of $t$, $a>0$ is a constant, $f:[0,a]\times E\times E\times E\rightarrow E$ is a Carath\'{e}odory type function, $u_0\in E$, $T$ and $S$ are Volterra and Fredholm integral operators, respectively. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the condition of uniformly continuity of the nonlinearity is not required, and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted. As samples of applications, we consider the initial value problem to a class of time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition at the end of this paper.

Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations

The main concern of this paper is to develop and analyze the high-order Runge–Kutta convolution quadrature (RKCQ) method for obtaining the numerical solution of nonlinear fractional integro-differential equations (FIDEs) with weakly singular kernels. We first study the existence and uniqueness of solutions for the original problem. Then, the convergence and stability results of the RKCQ method are obtained. Finally, some numerical experiments are reported to illustrate the effectiveness of the proposed schemes.

Existence of Solutions to Sobolev Type Nonlocal Nonlinear Functional Integrodifferential Equations Involving Caputo Derivative

In this article, we consider a nonlinear Sobolev type fractional functional integrodifferential equations in a Banach space along with a nonlocal condition. Sufficient conditions for existence, uniqueness and dependence on initial data of local solutions of considered problem are derived by employing fixed point techniques and theory of classical semigroup. Further, we also render the criteria for existence of global solution. At the end, we provide an application to elaborate the obtained results.