Integro-differential Evolution Equations Research Articles

A semi-analytical x-space solution for parton evolution — Application to non-singlet and singlet DGLAP equation

We present a novel semi-analytical method for parton evolution. It is based on constructing a family of analytic functions spanning x-space which is closed under the considered evolution equation. Using these functions as a basis, the original integro-differential evolution equation transforms into a system of coupled ordinary differential equations, which can be solved numerically by restriction to a suitably chosen finite subsystem. The evolved distributions are obtained as analytic functions in x with numerically obtained coefficients, providing insight into the analytic behavior of the evolved parton distributions. As a proof-of-principle, we apply our method to the leading order non-singlet and singlet DGLAP equation. Comparing our results to traditional Mellin-space methods, we find good agreement. The method is implemented in the code POMPOM in Mathematica as well as in Python.

Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions

Abstract In this article, we investigate the existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions. Using the theory of resolvent operators, some fixed point theorems, and an estimation technique of Kuratowski measure of noncompactness, we first establish some existence results of mild solutions for the proposed equation. Subsequently, we show by applying a newly established lemma that these solutions have regularity property under some conditions. Finally, as a sample of application, the obtained results are applied to a class of non-autonomous nonlocal partial integrodifferential equations.

Existence and Stability Results of Nonlinear Random Impulsive Integro-Differential Evolution Equations with Time-Varying Delays

This study examines the existence, uniqueness, and stability of nonlinear random impulsive integrodifferential equations with time-varying delays under sufficient conditions. Our study is based on the Leray-Schauder alternative fixed point theorem, Pachpatte’s inequality, and the Banach contraction principle. Besides, we generalize, extend, and develop some results for Kasinathan et al.

On the S-asymptotically ω-periodic mild solutions for multi-term time fractional measure integro-differential equations

This paper deals with a class of nonlocal problem for multi-term time-fractional measure integro-differential evolution equations of mixed type in Hilbert spaces. Firstly, we introduce the concept of S-asymptotically ω-periodic mild solution, on the premise of by utilizing -resolvent family and measure functional (Henstock–Lebesgue–Stieltjes integral). Secondly, combining regulated functions and fixed point theorem, the existence S-asymptotically ω-periodic mild solution for a classes of measure-driven differential equation is discussed. Finally, as samples of applications, we consider the nonlocal problem to a class of time-fractional measure partial differential equation with nonlocal condition at the end of this paper.

Asymptotically almost automorphy for impulsive integrodifferential evolution equations with infinite time delay via Mönch fixed point

Asymptotically almost automorphy for impulsive integrodifferential evolution equations with infinite time delay via Mönch fixed point

Approximate controllability for some integrodifferential evolution equations with nonlocal conditions

The main objective of this work is to investigate the existence of mild solutions and approximate controllability for some integrodifferential evolution equations with nonlocal conditions. Assuming that the linear part is exactly null and approximately controllable, using resolvent operator theory, we provide our main results. An example is given to illustrate the basic results of this work.

Principal spectral theory for nonlocal systems and applications to stem cell regeneration models

We first study the eigenvalue problem for a system of coupled integral operators and investigate existence and monotonicity of the principal eigenvalue and asymptotic behavior of the principal eigenpair. As applications of the spectral theory, we investigate a single genotype stem cell regeneration model with epigenetic transitions, and multiple genotype stem cell regeneration models with epigenetic transitions, with or without gene mutations. These models give rise to three classes of nonlinear integro-differential evolution equations. We consider existence, uniqueness and multiplicity of positive steady states to these equations, as well as the long time behavior of time-dependent solutions. Various explicit formulae for threshold values for tissue development, degeneration and abnormal growth are obtained.

Ulam–Hyers–Rassias Stability of Neutral Functional Integrodifferential Evolution Equations with Non-instantaneous Impulses on an Unbounded Interval

Ulam–Hyers–Rassias Stability of Neutral Functional Integrodifferential Evolution Equations with Non-instantaneous Impulses on an Unbounded Interval

Hasimoto transformation of general flows expressed in the Frenet frame

A one-dimensional space curve in R3 is a useful nonlinear medium for modeling vortex filaments and biological soft-matter capable of supporting a variety of wave motions. Hasimoto's transformation defines a mapping between the kinematic evolution of a space curve and nonlinear scalar equations evolving its intrinsic curve geometry. This mapping is quite robust and able to transform general vector fields expressed in the Frenet frame, resulting in a fully nonlinear integro-differential evolution equation, whose coefficient structure is defined by the coordinates of the flow in the Frenet frame. In this paper, we generalize the Hasimoto map to arbitrary flows defined on space curves, which we test against several existing kinematic flows. After this, we consider the time dynamics of length and bending energy to see that binormal flows are generally length preserving, and bending energy is fragile and unlikely to be conserved in the general case.

Optimal Control of Fractional Integro-differential Evolution Equation with Infinite Delay of order (1; 2)

In this paper, we will examine the facts in the image of infinite time delay on the mild solution’s solvability and fractional integro-differential Evolution Equation of Banach space’s optimal control choosing order ? ? (1,2). For investigation of mild solution of the system, continuation, uniqueness and the existence of the solution with the help of grown-wall inequality will be discussed shortly. After constructing the lagrange problem, the demonstration of actuality of optimal control of fractional integro delay system is being proceed. And at the end we will discuss the regarding result by an adequate example.

Existence and asymptotic properties of solutions of an integro-differential evolution equation with nonlocal conditions on infinite interval

Existence and asymptotic properties of solutions of an integro-differential evolution equation with nonlocal conditions on infinite interval

Existence and Differentiability of Solutions for Nondensely Defined Neutral Integro-Differential Evolution Equations

Existence and Differentiability of Solutions for Nondensely Defined Neutral Integro-Differential Evolution Equations

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.

Optimal control for Hilfer fractional neutral integrodifferential evolution equations with infinite delay

AbstractThe optimal control solutions for Hilfer fractional neutral integrodifferential evolution equations with infinite delay in Banach space are investigated in this study. To begin, we use fractional computations, Holder inequality, and Banach fixed point approaches to examine conclusions for fractional integrodifferential evolution equations. Following that, we created the fractional integrodifferential evolution system's continuous dependency. We also looked at optimal controls for the Lagrange problem. Finally, an application is given to extract the primary theory.

Mild Solutions for Fractional Impulsive Integro-Differential Evolution Equations with Nonlocal Conditions in Banach Spaces

In this paper, by using the cosine family theory, measure of non-compactness, the Mönch fixed point theorem and the method of estimate step by step, we establish the existence theorems of mild solutions for fractional impulsive integro-differential evolution equations of order 1<β≤2 with nonlocal conditions in Banach spaces under some weaker conditions. The results obtained herein generalizes and improves some known results. Finally, an example is presented for the demonstration of obtained results.

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.

Stochastic solutions of generalized time-fractional evolution equations

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Marichev-Saigo-Maeda time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Lévy motion for suitable pseudo-differential operators in space.

Existence and continuous dependence results for fractional evolution integrodifferential equations of order [formula omitted

The article analyzes the existence of Caputo fractional evolution integrodifferential equations of order 1<r<2 in Hilbert space with delay. A new set of adequate requirements for the existence outcomes of fractional delay evolution integrodifferential equations have been developed and are shown using the fractional derivative, Krasnoselskii’s fixed point theorem, and Henry-Gronwall inequalities. In addition, for the provided system, we developed continuous dependence results. Afterward, we apply our findings to the concept of nonlocal conditions. Then, to demonstrate our primary outcomes, two examples are given.

Existence and Asymptotic Periodicity of Solutions for Neutral Integro-Differential Evolution Equations with Infinite Delay

Abstract In this work, making use of the theory of resolvent operators and Banach fixed point theorem, we first discuss the existence and regularity of mild solutions for neutral partial functional integro-differential equations with infinite delay. We assume that the linear part of the considered equation generates a resolvent operator and the nonlinear function satisfies Lipschitz conditions. Then we investigate the asymptotic periodicity of mild solutions under asymptotic periodic assumption on the nonlinear function. The obtained results extend somewhat the related conclusions in literature. In the end, an example is presented to illustrate the obtained results.

A discussion on boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations

Sufficient conditions for boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations are explored. In the first problem, the outcomes are proven with the help of Sadovskii's fixed point theorem collaborated with strongly continuous semigroup theory. In the second problem, we considered nonautonomous evolution equations, and the result is proven by employing Sadovskii's fixed point theorem in collaboration with the evolution system. Examples are included to demonstrate the theoretical results for both types of equations.