Nonlocal Evolution Equations Research Articles

Existence results of mild solutions for nonlocal fractional delay integro-differential evolution equations via Caputo conformable fractional derivative

<abstract><p>In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauchy problem with Caputo conformable derivative in Banach spaces. We establish a representation of a mild solution using a fractional Laplace transform. The existence of such solutions is proved under certain conditions, using the Mönch fixed point theorem and a general version of Gronwall's inequality under weaker conditions in the sense of Kuratowski measure of non compactness. Applications illustrating our main abstract results and showing the applicability of the presented theory are also given.</p></abstract>

Existence of mild solutions for nonlocal ψ−Caputo-type fractional evolution equations with nondense domain

Abstract The main crux of this manuscript is to establish the existence and uniqueness of solutions for nonlocal fractional evolution equations involving ψ−Caputo fractional derivatives of an arbitrary order α ∈ (0, 1) with nondense domain. The mild solutions of our proposed model are constructed by employing generalized ψ−Laplace transform and some new density functions. The proofs are based on Krasnoselskii fixed point theorem and some basic techniques of ψ−fractional calculus. As application, a nontrivial example is given to illustrate our theoritical results.

Nonlocal fuzzy fractional stochastic evolution equations with fractional Brownian motion of order (1,2)

<abstract><p>In this manuscript, we formulate the system of fuzzy stochastic fractional evolution equations (FSFEEs) driven by fractional Brownian motion. We find the results about the existence-uniqueness of the formulated system by using the Lipschitizian conditions. By using these conditions we have also investigated the exponential stability of the solution for the above system driven by fractional Brownian motion. Finally, the applications in financial mathematics are presented and the use of financial mathematics in the fractional Black and Scholes model is also discussed. An example is propounded to show the applicability of our results.</p></abstract>

On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

<p style='text-indent:20px;'>We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.</p>

Fast ADI difference/compact difference schemes for the nonlocal evolution equation with weakly singular kernels in three dimensions

We consider two kinds of efficient numerical methods to solve the three-dimensional evolution equation with weakly singular kernels. The proposed techniques are based on the finite difference/compact difference methods for the spatial discretization and an alternating direction implicit (ADI) algorithm for the time direction, combined with the second-order convolution quadrature (CQ) rule for the Riemann–Liouville (R-L) integral term and the classical L1 formula for the Caputo fractional derivative. The unconditional stability and convergence of the ADI finite difference scheme are proved rigorously. Numerical results are presented to support the theoretical analysis.

On the limit of a two‐phase flow problem in thin porous media domains of Brinkman type

We study the process of two‐phase flow in thin porous media domains of Brinkman type. This is generally described by a model of coupled, mixed‐type differential equations of fluids' saturation and pressure. To reduce the model complexity, different approaches that utilize the thin geometry of the domain have been suggested. We focus on a reduced model that is formulated as a single nonlocal evolution equation of saturation. It is derived by applying standard asymptotic analysis to the dimensionless coupled model; however, a rigid mathematical derivation is still lacking. In this paper, we prove that the reduced model is the analytical limit of the coupled two‐phase flow model as the geometrical parameter of domain's width–length ratio tends to zero. Precisely, we prove the convergence of weak solutions for the coupled model to a weak solution for the reduced model as the geometrical parameter approaches zero.

A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem

This work constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space. In the temporal direction, the Crank-Nicolson (CN) method and product-integration (PI) rule are employed, from which the non-uniform meshes are used to eliminate the singular behaviour of the exact solution at t=0. Then, a fully discrete scheme is obtained by the spatial discretization based on the finite difference method. Simultaneously, an alternating direction implicit (ADI) algorithm is designed to reduce the computational cost. The stability in L2 norm and convergence are derived via the energy method, in which the convergence orders are O(k2+h2) with certain suitable assumptions, where k and h are corresponding space-time step sizes, respectively. Numerical results confirm the theoretical analysis.

Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows

This article studies a coupled system of model multi-dimensional partial differential equations (PDEs) that arise in the nonlinear dynamics of two-fluid Couette flow when insoluble surfactants are present on the interface. The equations have been derived previously, but a rigorous study of local and global existence of their solutions, or indeed solutions of analogous systems, has not been considered previously. The evolution PDEs are two-dimensional in space and contain novel pseudo-differential terms that emerge from asymptotic analysis and matching in the multi-scale problem at hand. The one-dimensional surfactant-free case was studied previously, where travelling wave solutions were constructed numerically and their stability investigated; in addition, the travelling wave solutions were justified mathematically. The present study is concerned with some rigorous results of the multi-dimensional surfactant system, including local well posedness and smoothing results when there is full coupling between surfactant dynamics and interfacial motion, and global existence results when such coupling is absent. As far as we know such results are new for non-local thin film equations in either one or two dimensions.

Large strain integral-based nonlocal simulation of ductile damage with application to mode-I fracture

The integral approach to nonlocal ductile damage is developed to enable numerically stable and physically reasonable simulations of crack initiation and propagation in metals. A previously proposed phenomenological model of finite strain plasticity is delocalized through introducing a nonlocal evolution equation for porosity. In the new model, the strain localization is controlled by at least one internal length parameter, thus avoiding unphysical strain localization into a zero thickness layer. For numerical tests, meshless smoothed particle simulations of crack propagation in a compact tension specimen are carried out. The new nonlocal model’s performance is analysed in terms of the force–displacement curves, critical stress intensity factors, energy dissipation during crack propagation, and geometry of the damaged zone. The impact of the delocalization procedure on structural strength under mode-I fracture is clarified: (i) isotropic and anisotropic delocalization procedures are considered; (ii) the averaging operator is applied both on the reference and current configurations; (iii) damage–continuity interaction is introduced to reduce unphysical spreading of damage.

Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

In this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.

Pattern formation in nonlocal Kondo model

We study a nonlocal evolution equation generalising a model introduced by Shigeru Kondo to explain colour patterns on a skin of a guppy fish. We prove the existence of stationary solutions using either the bifurcation theory or the Schauder fixed‐point theorem. We also present numerical studies of this model and show that it exhibits patterns similar to those modelled by well‐known reaction‐diffusion equations.

A hierarchy of nonlocal nonlinear evolution equations and [formula omitted]-dressing method

Starting from the matrix ∂̄ problem, the dressing method is employed to investigate the nonlocal nonlinear evolution equations. A hierarchy of nonlocal nonlinear evolution equations associated with 2 × 2 matrix problem, which contains the nonlocal extended modified Korteweg–de Vries (emKdV) equation, is derived via using recursive operator for the first time. Finally, via selecting a proper spectral transformation matrix, the N-soliton solutions of the nonlocal emKdV equation are constructed based on the matrix ∂̄ problem.

On a class of nonlocal evolution equations with the p[∇u]-Laplace operator

We study the homogeneous Dirichlet problem for the class of singular parabolic equationsut−div(|∇u|p[∇u]−2∇u)=fin Ω×(0,T), where Ω⊂Rd, d≥2, is a smooth domain. The exponent p nonlocally depends on the gradient of the solution: p is a given function defined byp[∇u]≡p(l(|∇u|)),l(|s|)=∫Ω|s|αdx with a constant α∈(1,2]. We find sufficient conditions on the data that guarantee global in time existence and uniqueness of a strong solution of the problem. It is shown that the problem has a solution if either u0 and f, or p′(s) are sufficiently small.

Controllability for nonlocal stochastic integrodifferential evolution equations with the lack of compactness

In this work, we study the controllability of a class of stochastic integrodifferential equations with noncompact semigroups and nonlocal condition on Hilbert spaces.We suppose that the linear part admits a resolvent operator in the sense of Grimmer. Our main results are obtained by utilizing stochastic analysis theory, Kuratowski measure of noncompactness and Mönch fixed point Theorem. As a result, we obtain a generalization of a host of important results in the literature, without assuming the compactness of the resolvent operator. Finally, we provide an example to illustrate the efficiency of our results.

Optimal control of nonlocal fractional evolution equations in the \u03b1-norm of order (1,2)

This paper investigates the optimal control for a class of nonlocal fractional evolution equations of order gamma in (1,2) in Banach spaces. An adequate definition of α-mild solutions is obtained and the existence, uniqueness and continuous dependence of α-mild solutions for the presented control system are also established. The existence of optimal pairs of nonlocal fractional evolution systems is also demonstrated with a view on the construction of the Lagrange problem. Finally, an example is propounded for the presentation of optimal control.

Mild solutions of nonlocal semilinear evolution equations on unbounded intervals via approximation solvability method in reflexive Banach spaces

We study the existence of mild solutions for a nonlocal semilinear evolution equation on unbounded interval by means of an approximation solvability method without assuming compactness on the evolution system and on the nonlinearity. The method is based on the reduction to a finite dimensional problem by means of the projections solved by a fixed point approach that includes a compactness criterion on BC([0,+∞),E). Continuation principle and weak topology are used as well.

Spectral Gap for the Growth-Fragmentation Equation via Harris's Theorem

We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalise those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the non-conservativeness of the equation is overcome by performing an $h$-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.

Nonlocal and nonlinear evolution equations in perforated domains

In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form ut(x,t)=∫J(x−y)u(y,t)dy−hε(x)u(x,t)+f(x,u(x,t)) with x in a perturbed domain Ωε⊂Ω which is thought as a fixed set Ω from where we remove a subset Aε called the holes. We choose appropriated families of functions hε∈L∞ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside Ω. Moreover, we take J as a non-singular kernel and f as a nonlocal nonlinearity. Under the assumption that the characteristic functions of Ωε have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation.

Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length

We study local (the heat equation) and nonlocal (convolution-type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star-shaped graph in which there are only one node and as many infinite edges as in the original graph. In this way, we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. For this star-shaped limit problem, the asymptotic behaviour of the solutions is just given by the solution to the heat equation in a half line with a Neumann boundary condition at $$x=0$$ and initial datum $$(2 M/N ) \delta _{x=0}$$ where M is the total mass of the initial condition for our original problem and N is the number of edges of infinite length. In addition, we show that solutions to the nonlocal problem converge, when we rescale the kernel, to solutions to the heat equation (the local problem), that is, we find a relaxation limit.

A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices

In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta–Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model.