Nonlinear Fractional Integro-differential Equations Research Articles

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.

Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fr&amp;#233;chet spaces

This paper deals with the existence of mild solutions for non-linear fractional integrodifferential equations with state-dependent nonlocal conditions. The technique used is a generalization of the classical Darbo fixed point theorem for Frechet spaces associated with the concept of measures ′ of noncompactness. An application of the main result has been included.

Approximate solution of nonlinear fractional integro-differential equations using fractional alternative Legendre functions

In this paper, a new set of functions called fractional alternative Legendre is defined for solving nonlinear fractional integro-differential equations. The concept of the fractional derivative in this problem is in the Caputo sense. To solve the problem, first we obtain the operational matrix of the Riemann–Liouville fractional integration of these functions for the first time. Then, this matrix and collocation method are utilized to reduce the solution of the nonlinear fractional integro-differential equations to a system of algebraic equations. Also, the convergence analysis of the proposed method is investigated. Finally, some examples are included to demonstrate the validity and applicability of the approach.

A discrete orthogonal polynomials approach for coupled systems of nonlinear fractional order integro-differential equations

This paper develops a numerical approach for solving coupled systems of nonlinear fractional order integro-differential equations(NFIDE). Shifted discrete Chebyshev polynomials (SDCPs) have been introduced and their attributes have been checked. Fractional operational matrices for the orthogonal polynomials are also acquired. A numerical algorithm supported by the discrete orthogonal polynomials and operational matrices are used to approximate solution of coupled systems of NFIDE. The operational matrices of fractional integration and product are applied for approximate the unknown functions directly. These approximations were put in the coupled systems of NFIDE. A comparison has been made between the absolute error of approximate solutions of SDCPs method with previous published. The gained numerical conclusions disclose that utilizing discrete Chebyshev polynomials are more efficient in comparison to the other methods.

ON FRACTIONAL VOLTERRA INTEGRODIFFERENTIAL EQUATIONS WITH FRACTIONAL INTEGRABLE IMPULSES

We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki’s normed Banach spaces. Further, Bielecki-Ulam type stabilities have been demonstrated on a compact interval. A concrete example is provided to illustrate the outcomes we acquired.

Existence and uniqueness of the global solution for a class of nonlinear fractional integro-differential equations in a Banach space

In this paper, by employing fixed point theory, we investigate the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations on semi-infinite domains in a Banach space.

On Nonlinear Fractional Integro–Differential Equations with Positive Constant Coefficient

The aim of this study is to investigate the existence and other properties of solution of nonlinear fractional integro–differential equations with constant coefficient. Also with the help of Pachpatte’s inequality, we prove the continuous dependence of the solutions.

Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations

This paper considers the nonlinear stochastic fractional integro-differential equations (SFIDEs) under the non-Lipschitz conditions, which are general and include many stochastic (fractional) integro-differential equations discussed in literature. An important connection between SFIDEs and stochastic Volterra integral equations (SVIEs) is derived in detail by the Fubini theorem. Using the Euler–Maruyama (EM) approximation, we prove the existence, uniqueness and stability results of the solution to SFIDEs. Moreover, it is shown that the modified EM solution of SFIDEs shares strong first-order sharp convergence. The numerical examples are performed to show the accuracy and effectiveness of the numerical scheme and verify the correctness of our theoretical analysis. • The model covers many nonlinear stochastic (fractional) integro-differential models. • Well-posedness of the models is investigated under the non-Lipschitz condition. • The modified EM approximation of the models admits strong first-order convergence.

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

Approximating Solutions of Nonlinear Hybrid Caputo Fractional Integro-Differential Equations via Dhage Iteration Principle

In this article, we prove the existence and approximation of solutions of the initial value problems of nonlinear hybrid Caputo fractional integro-differential equations. The main tool employed here is the Dhage iteration principle in a partially ordered normed linear space. An example is also given to illustrate the main results.

On nonlinear mixed fractional integro-differential equations with positive constant coefficient

In this paper, we study the existence and other properties of the solution of nonlinear mixed fractional integro-differential equations with constant coefficient. Also with the help of integral inequality of mixed type, we prove the continuous dependence of the solutions on the initial conditions.

On the weak solutions of an overdetermined system of nonlinear fractional partial integro-differential equations

On the weak solutions of an overdetermined system of nonlinear fractional partial integro-differential equations

Existence and uniqueness of solutions for nonlinear fractional integrodifferential equations with non-local boundary conditions

Existence and uniqueness of solutions for nonlinear fractional integrodifferential equations with non-local boundary conditions

Reproducing Kernel Method for Solving Nonlinear Fractional Fredholm Integrodifferential Equation

This article is devoted to both theoretical and numerical studies of nonlinear fractional Fredholm integrodifferential equations. In this paper, we implement the reproducing kernel method (RKM) to approximate the solution of nonlinear fractional Fredholm integrodifferential equations. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the solution of the nonlinear fractional Fredholm integrodifferential equation. Uniformly convergence of the approximate solution produced by the RKM to the exact solution is proven.

Existence and exponential behavior of multi-valued nonlinear fractional stochastic integro-differential equations with Poisson jumps of Clarke’s subdifferential type

This manuscript addresses the study of a new class of multi-valued nonlinear fractional stochastic integro-differential equations with Poisson jumps of Clarke’s subdifferential type in Hilbert space of order 1<α<2. Initially, by using α-resolvent operator, Holder inequality, properties of Clarke’s generalized subdifferential, fractional calculus and the multi-valued fixed point theorem due to Dhage, the existence result for the proposed system is obtained. Further, the sufficient conditions are established to ensure that exponential decay of mild solution to zero in the square mean. Finally, the obtained results are applied to fractional stochastic hemivariational inequalities. An example is illustrated for the development of obtained results.

Existence of Solutions for Fractional Integro-Differential Equations with Non-Local Boundary Conditions

In this paper, we study the existence of solutions for a new class of boundary value problems of non-linear fractional integro-differential equations. The existence result is obtained with the aid of Schauder type fixed point theorem while the uniqueness of solution is established by means of contraction mapping principle. Then, we present some examples to illustrate our results.

A sinc-Gauss-Jacobi collocation method for solving Volterra's population growth model with fractional order

A new sinc-Gauss-Jacobi collocation method for solving the fractional Volterra's population growth model in a closed system is proposed. This model is a nonlinear fractional Volterra integro-differential equation where the integral term represents the effects of toxin. The fractional derivative is considered in the Liouville-Caputo sense. In the proposed method, we first convert fractional Volterra's population model to an equivalent nonlinear fractional differential equation, and then the resulting problem is solved using collocation method. The proposed collocation technique is based on sinc functions and Gauss-Jacobi quadrature rule. In this approach, the problem is reduced to a set of algebraic equations. The obtained numerical results of the present method are compared with some well-known results in the literature to show the applicability and efficiency of the proposed method.