In the present paper initial problem for the integro-differential diffusion system with nonlocal nonlinear source is considered. The results on existence of local mild solution and nonexistence of global weak solution to the nonlinear integro-differential diffusion system are presented.

This paper investigates the boundary finite-time stabilization of fractional reaction-diffusion systems (FRDSs). First, a distributed controller is designed, and sufficient conditions are obtained to ensure the finite-time stability (FTS) of FRDSs under the designed controller. Then, a boundary controller is presented to achieve the FTS. By virtue of Lyapunov functional method and inequality techniques, sufficient conditions are presented to ensure the FTS of FRDSs via the designed boundary controller. The effect of diffusion term of FRDSs on the FTS is also investigated. Both Neumann and mixed boundary conditions are considered. Moreover, the robust finite-time stabilization of uncertain FRDSs is studied when there are uncertainties in the system’s coefficients. Under the designed boundary controller, sufficient conditions are presented to guarantee the robust FTS of uncertain FRDSs. Finally, numerical examples are presented to verify the effectiveness of our theoretical results.

It is established $L^{p}$ estimates for the fractional $\Phi$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the Moser’s iteration, we prove that any weak solution for fractional $\Phi$-Laplacian operator defined in bounded domains belongs to $L^\infty(\Omega)$ under appropriate hypotheses on the $N$-function $\Phi$. Using the Orlicz space and taking into account the fractional setting for our problem the main results are stated for a huge class of nonlinear operators and nonlinearities.

We derive the fractional version of one-phase one-dimensional Stefan model. We assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model. Furthermore, we find a special solution to this problem.

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplicity, such an equation can describe an amazing number of dynamical behaviours. In this paper, we shall provide a comprehensive survey of possible bifurcation patterns, deriving explicit conditions on the equation's parameters for the occurrence of each of them, and discuss illustrative examples.

In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials ( P α ). As a consequence of energy argument, we derive L ∞ ( 0 , T ; H 0 1 ( Ω ) ) bound as well as L 2 ( 0 , T ; H 2 ( Ω ) ) bound on the solution of the problem ( P α ) by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem is established. Further, we study semi discrete formulation of the problem ( P α ) by discretizing the space domain using a conforming FEM and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra projection operator in such a way that it reduces the complexities arising from the Kichhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem ( P α ) with a convergence rate of O ( h + k 2 − α ) , where α (0 1) is the fractional derivative exponent, h and k are the discretization parameters in the space and time directions respectively. This convergence rate is improved to second order in the time direction by proposing a novel linearized L2-1 σ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims.

An initial-boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann-Dirichlet type is investigated in this work. From a physical point of view, the initial-boundary value problem considered here is a mathematical model of quasi-stationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. In this article, by the Galerkin method to prove the existence of a weak solution to the initial-boundary value problem for a pseudoparabolic equation in a bounded domain. On the basis of a priori estimates, we prove a local existence theoremand uniqueness for a weak generalized solution of the initial-boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann-Dirichlet boundary condition are obtained.

We study the ground state solutions for the following p\&q-Laplacain equation \[ \left\{ \begin{array}{ll} -\Delta_pu-\Delta_qu+V(x) (|u|^{p-2}u+|u|^{q-2}u)=\lambda K(x)f(u)+|u|^{q^*-2}u,~x\in\R^N, \\ u\in W^{1,p}(\R^N)\cap W^{1,q}(\R^N), \end{array} \right. \] where $\lambda>0$ is a parameter large enough, $\Delta_ru = \text{div}(|\nabla u|^{r-2}\nabla u)$ with $r\in\{p,q\}$ denotes the $r$ Laplacian operator, $1

In this paper, we introduce two simple inertial algorithms for solving the split variational inclusion problem in Banach spaces. Under mild and standard assumptions we establish the weak and strong convergence of the proposed methods, respectively. As theoretical realization we study existence of solutions of the split common fixed point problem in Banach spaces. Several numerical examples in finite and infinite dimensional spaces compare and illustrate the performances of our schemes. Our work generalize and extend some recent relate results in the literature and also propose a simple and applicable method for solving split variational inclusions. .2in _2000 Mathematics Subject Classification:_ 65K10; 65K05; 47H10; 47L25. _Keywords:_ Split variational inclusion problem; inertial technique; Banach spaces.

A linear manifold ${\mathcal K}_2$ of evolutionary equations for a pseudovector field on ${\Bbb R}^3$ is described. An infinitisimal shift of each equation is determined by a second-order differential operator of divergent type. All operators are invariant with respect to space translations in ${\Bbb R}^3$, relative to time translations, and they are transformed by covariant way relative to rotations of ${\Bbb R}^3$. It is proved that the linear space ${\mathcal M}_2 \subset {\mathcal K}_2$ of differential operators preserving solenoidal property and unimodularity of the field is one-dimensional and an explicit form of such operators is found.

In this manuscript,we present a parametric family of derivative-free 3-steps iterative methods with a weight function for solving nonlinear equations. We study various ways of introducing memory to this parametric family in order to increase the order of convergence without new functional evaluations. We also performed numerical experiments to compare the iterative methods fromdifferent points of view.

We apply the Bielecki metric on the space C([a, b]), to analyze the different types of stabilities of non-linear fractional integral equation corresponding to fractional boundary value problems. Sufficient conditions are obtained to prove stability results for fractional non-linear Volterra and Fredholm integral equations, given by Ulam, Hyer and Rassias. We extend the respective stability results to the fractional integral equations where the domain of integration is an unbounded interval. We provide numerical examples which asserts our stability results.

The aim of this paper is to construct a numerical scheme for solving a class of bilateral free boundaries problem. First, using a shape functional and some regularization terms, an optimal control problem is formulated, in addition, we prove its solution existence's. The first optimality conditions and the shape gradient are computed. the proposed numerical scheme is a genetic algorithm guided conjugate gradient combined with the finite element method, at each mesh regeneration, we perform a mesh refinement in order to avoid any domain singularities. Some numerical examples are shown to demonstrate the validity of the theoretical results, and to prove the robustness of the proposed scheme.

The paper deals with the analysis of stable thermo-solutal dendritic growth in the presence of intense convection. The n-fold symmetry of crystalline anisotropy as well as the two- and three-dimensional growth geometries are considered. The steady-state analytical solutions are found with allowance for the convective-type heat and mass exchange boundary conditions at the dendritic surface. A linear morphological stability analysis determining the marginal wavenumber is carried out. The new stability criterion is derived from the solvability theory and stability analysis. This selection criterion takes place in the regions of small undercooling. To describe a broader undercooling diapason, the obtained selection criterion, which describes the case of intense convection, is stitched together with the previously known selection criterion for the conductive-type boundary conditions. It is demonstrated that the stitched selection criterion well describes a broad diapason of experimental undercoolings.

In this paper, we adopt the optimize-then-discretize approach to solve parabolic optimal Dirichlet boundary control problem. First, we derive the first-order necessary optimality system, which includes the state, co-state equations and the optimality condition. Then, we propose Crank-Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible and second-order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present a numerical example to validate our theoretical analysis.

A centroid- and covariance-invariant deterministic mapping of sets of discrete data points to nonlinear models is introduced. Conditions for bijectivity of this mapping are developed. Since the mapping can be accomplished by look-up tables for the special case of equally-spaced data, the resulting mapping algorithm is considered computationally fast. This could be attractive for real-time operations.

This paper aims to solve the celebrated Fuzzy Fractional Differential Equations (FFDE) using an Artificial Neural Network (ANN) technique. Compared to the integer order differential equation, the proposed FFDE can better describe several real application problems of various physical systems. To accomplish the aforementioned aim, the error back propagation algorithm and a multi-layer feed forward neural architecture are utilized using the unsupervised learning in order to minimize the error function as well as the modification of the parameters such as weights and biases. By combining the initial conditions with the ANN, output provides an appropriate approximate solution of the proposed FFDE. Then, two illustrative examples are solved to confirm the applicability of the concept as well as to demonstrate both the precision and effectiveness of the developed method. By comparing with some traditional methods, the obtained results reveals a close match that confirms both accuracy and correctness of the proposed method.

This paper is concerned with the time-decay rates of the strong solutions of the three dimensional non-isentropic compressible magnetohydrodynamic (MHD) system. First, motivated by Pu–Guo’s result [Z. Angew. Math. Phys. 64 (2013) 519–538], we establish the existence result of a unique local-in-time strong solution for the MHD system. Then, we derive a priori estimates and use the continuity argument to obtain the global-in-time solution, where the initial data should be bounded in L1-norm and is small in H2-norm. Finally, based on Fourier theory and the idea of cancellation of a low-medium frequent part as in [Sci. China Math. 65 (2022) 1199–1228], we get the optimal time-decay rates (including highest-order derivatives) of strong solutions for non-isentropic MHD fluids. Our result is the first one concerning with the optimal decay estimates of the highest-order derivatives of the non-isentropic MHD system.