Semilinear Integro-differential Equations Research Articles

Semilinear multi-term fractional in time diffusion with memory

In this study, the initial-boundary value problems to semilinear integro-differential equations with multi-term fractional Caputo derivatives are analyzed. A particular case of these equations models oxygen diffusion through capillaries. Under proper requirements on the given data in the model, the classical and strong solvability of these problems for any finite time interval [0, T] are proved via so-called continuation method. The key point in this approach is finding suitable a priori estimates of a solution in the fractional Hölder and Sobolev spaces.

(ω,c)-Periodic Solution to Semilinear Integro-Differential Equations with Hadamard Derivatives

(ω,c)-Periodic Solution to Semilinear Integro-Differential Equations with Hadamard Derivatives

Controllability of Hilfer Fractional Semilinear Integro-Differential Equation of Order α ∊ (0, 1), β ∊ [0, 1

In this paper, Hilfer fractional differential equation is studied. Firstly, we used Laplace transform and semigroup theory to find the mild solution of the system. Then exact controllability of proposed system is established using the Arzela-Ascoli theorem and Schauder fixed point theorem. To illustrate the developed theory we provide an example at the end.

A Uniformly Convergent Numerical Method for Singularly Perturbed Semilinear Integro-Differential Equations with Two Integral Boundary Conditions

A Uniformly Convergent Numerical Method for Singularly Perturbed Semilinear Integro-Differential Equations with Two Integral Boundary Conditions

Explicit Exponential Runge–Kutta Methods for Semilinear Integro-Differential Equations

Explicit Exponential Runge–Kutta Methods for Semilinear Integro-Differential Equations

Existence and Compactness of Solutions of Semilinear Integrodifferential Equations with Finite Delay

Existence and Compactness of Solutions of Semilinear Integrodifferential Equations with Finite Delay

Existence and compactness of solution of semilinear integro-differential equations with finite delay

UDC 517.9 We present some existence and uniqueness results for a class of functional integro-differential evolution equations generated by the resolvent operator for which the semigroup is not necessarily compact. It is proved that the set of solutions is compact. Our approach is based on fixed point theory. Finally, some examples are given to illustrate the results.

Functional Integro-Differential Equations with State-Dependent Delay and Non-Instantaneous Impulsions: Existence and Qualitative Results

This paper addresses some existence, attractivity and controllability results for semilinear integrodifferential equations having non-instantaneous impulsions on an infinite interval via resolvent operators in case of neutral and state-dependent delay problems. Our criteria were obtained by applying a Darbo’s fixed-point theorem combined with measures of noncompactness. The obtained result is illustrated by an example at the end.

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

Optimal control for semilinear integrodifferential evolution equations in Banach spaces

In this paper, the optimal controls for semilinear integrodifferential evolution equations are investigated in Banach spaces. We demonstrate the existence of feasible pairs by utilising the Cesari condition and the Filippov Theorem. Furthermore, a result on the existence of optimal control pairs for the Lagrange issue is presented.

A Note on Approximate Controllability of Fractional Semilinear Integrodifferential Control Systems via Resolvent Operators

This article primarily focuses on the approximate controllability of fractional semilinear integrodifferential equations using resolvent operators. Two alternative sets of necessary requirements have been studied. In the first set, we use theories from functional analysis, the compactness of an associated resolvent operator, for our discussion. The primary discussion is proved in the second set by employing Gronwall’s inequality, which prevents the need for compactness of the resolvent operator and the standard fixed point theorems. Then, we extend the discussions to the fractional Sobolev-type semilinear integrodifferential systems. Finally, some theoretical and practical examples are provided to illustrate the obtained theoretical results.

Existence results for Atangana-Baleanu fractional neutral integro-differential systems with infinite delay through sectorial operators

This manuscript’s main objective is to examine the existence of mild solution of Atangana-Baleanu fractional order semi-linear integro-differential equations [ABFSIDE] and semi-linear neutral integro-differential evolution equations [ABFSNIDE] with infinite delay [ID] in Banach spaces. We introduce an appropriate definition of a mild solution for these systems. Based on the Banach contraction principle, nonlinear alternative of Leray-Schauder type and Krasnoselskii-Schaefer fixed point theorem joined with ρ-resolvent operators, we develop the main results. Finally, an example is given to justify the theoretical results.

Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

Local and blowing-up solutions for an integro-differential diffusion equation and system

In the present paper initial problems for the semilinear integro-differential diffusion equation and system are considered. The analogue of Duhamel principle for the linear integro-differential diffusion equation is proved. The results on existence of local mild solutions and Fujita-type critical exponents to the semilinear integro-differential diffusion equation and system are presented.

Existence Results for Fractional Semilinear Integrodifferential Equations of Mixed Type with Delay

In this paper, we discuss a class of fractional semilinear integrodifferential equations of mixed type with delay. Based on the theories of resolvent operators, the measure of noncompactness, and the fixed point theorems, we establish the existence and uniqueness of global mild solutions for the equations. An example is provided to illustrate the application of our main results.

Mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions

In this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order \(\beta\in(1,2]\) subjected to noncompact initial nonlocal conditions. We assume that the linear part generates an arbitrarily strongly continuous \(\beta\)-order fractional cosine family, while the nonlinear forcing term is of Caratheodory type and satisfies some fairly general growth conditions. Our approach combines the Monch fixed point theorem with some recent results regarding the measure of noncompactness of integral operators. Our conclusions improve and generalize many earlier related works. An example is provided to illustrate the main results.

Two-grid scheme of expanded mixed finite element method for semilinear parabolic integro-differential equations

ABSTRACT In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog scheme at the rest time levels. Next, we make use of the known coarse mesh solution and Taylor expansion to infer the solution on the fine mesh space. Thus, we only need to solve the nonlinear problem once on the coarse grid space of the two-grid scheme. Finally, a numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation

This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in "narrow" sets. The existence of the solution was already proved in part I of this work.

From the Peierls–Nabarro model to the equation of motion of the dislocation continuum

We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls–Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a well known equation called by Head (1972) ”the equation of motion of the dislocation continuum”. The limit equation is a model for the macroscopic crystal plasticity with density of dislocations. In particular, we recover the so called Orowan’s law which states that dislocations move at a velocity proportional to the effective stress.

Existence Theorems for Fractional Semilinear Integrodifferential Equations with Noninstantaneous Impulses and Delay

In this paper, we consider a class of fractional semilinear integrodifferential equations with noninstantaneous impulses and delay. By the semigroup theory and fixed point theorems, we establish various theorems for the existence of mild solutions for the problem. An example involving partial differential equations with noninstantaneous impulses is given to show the application of our main results.