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An inverse problem of determining the right-hand side of the abstract subdiffusion equation with a fractional Caputo derivative is considered in a Hilbert space H. For the forward problem, instead of the Cauchy condition, the non-local in time condition u (0) = u (T) is taken. The right-hand side of the equation has the form g (t) f with a given function g (t) and an unknown element f ∈ H . If the function g (t) preserves its sign, then under a over-determined condition u (t 0) = ψ , t 0 ∈ (0 , T) , it is proved that the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution for some sign-changing functions g (t) . For such functions g (t) , under certain conditions on this function, one can achieve the well-posedness of the problem by choosing properly t 0 . Moreover, we show that for some g (t) , for the existence of a solution to the inverse problem, certain orthogonality conditions must be satisfied, but in this case there is no uniqueness. To the best knowledge of authors, the inverse problem with the non-local condition u (0) = u (T) has been considered for the first time. Moreover, all the results obtained are new not only for the subdiffusion equation, but also for the classical diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper, a problem for stochastic elliptic equations containing Wiener process and non-local condition is considered. Initially, existence results are constructed under two different hypotheses for the input data. Next, the instability behavior of the mild solution is presented, which shows that the solution does not depend continuously on the given data. After that, a regularized solution is established and its convergence rate is described. Lastly, an illustrated example is shown to test the proposed method. Our article was inspired by recent papers by Tomás Caraballo and his colleagues on the ill-posed problem for stochastic PDEs. [ABSTRACT FROM AUTHOR]

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented.

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on a control system with a non-local condition that is governed by a Hilfer fractional neutral stochastic evolution hemivariational inequality (HFNSEHVI). By using concepts of the generalized Clarke sub-differential and a fixed point theorem for multivalued maps, we first demonstrate adequate requirements for the existence of mild solutions to the concerned control system. Then, using limited Lagrange optimal systems, we demonstrate the existence of optimal state-control pairs that are regulated by an HFNSEHVI with a non-local condition. In order to demonstrate the existence of fixed points, the symmetric structure of the spaces and operators that we create is essential. Without considering the uniqueness of the control system's solutions, the best control results are established. Lastly, an illustration is used to demonstrate the major result. [ABSTRACT FROM AUTHOR]

This article deals with existence, uniqueness and Ulam–Hyers–Rassias stability solutions for complex order Hilfer-Katugampola type pantograph equations involving initial and nonlocal condition.

The study of equations of mathematical physics, including inverse problems, is relevant today. This work is devoted to the fundamental problem of studying the solvability and qualitative properties of the solution of the inverse problem for a quasilinear pseudoparabolic equation (also called Sobolev-type equations) with memory term. To date, studies of direct and inverse problems for a pseudoparabolic equations are rapidly developing in connection with the needs of modeling and control of processes in thermal physics, hydrodynamics, and mechanics of a continuous medium. The pseudoparabolic equations similar to those considered in this work arise in the description of heat and mass transfer processes, processes of non-Newtonian fluids motion, wave processes, and in many other areas. The main types of the inverse problems are: boundary, retrospective, coefficient and geometric. The boundary and retrospective inverse problems lead to the study of linear problems. In turn, the statements related to the study of coefficient and geometric types bring to the nonlinear problems. Coefficient inverse problems are divided into two main groups: coefficient inverse problems, where the unknown is a function of one or several variables, and finite-dimensional coefficient inverse problems. In this article the existence and uniqueness of a weak and strong solution of the inverse problem in a bounded domain are proved by the Galerkin method. Also we used Sobolev’s embedding theorems, and obtained a priori estimates for the solution. Moreover, we get local and global theorems on the existence of the solution.

We study linear backward stochastic partial differential equations (BSPDEs) of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution over time. We establish well-posedness, existence, uniqueness, and regularity, for the solutions of this new problem. In particular, this can be considered as a possibility to recover a solution of a BSPDE in a setting where its values at the terminal time are unknown, and where the average of the solution over time is preselected. [ABSTRACT FROM AUTHOR]

The purpose of this work is to use a new numerical technique for solving the two-sided multi-dimensional variable order fractional mobile/immobile diffusion equation with non-local conditions (TSMDVOF-MIDENLCs) model using the variable time fractional derivative of Caputo, as well as an initial boundary value problem of modified treatment. We used the fractional variational iteration method (FVIM) to mix initial and boundary conditions, resulting in for each iteration, a new initial solution. Convergence, sufficient conditions (SC) for system convergence, and error estimation are discussed. Some examples are given to illustrate the applicability of the novel suggested method, demonstrating that the numerical solution matches the exact solution and that the error is zero. Furthermore, this algorithm is easy and inexpensive to implement, and it demonstrates efficiency and accuracy. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematics, Mechanics & Computer Science is the property of Al-Farabi Kazakh National University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider a non-local boundary-value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional derivative in rectangular domain. The main target of this work is to analyze the uniqueness and the existence of the solution of the considered problem by means of eigenfunctions. Moreover, we construct the solution of the ordinary fractional differential equation with the rightsided bi-ordinal Hilfer derivative by the method of reduction to the Volterra integral equation. Then, we present sufficient conditions for given data in order to show the existence of the solution. [ABSTRACT FROM AUTHOR]

In the work, a boundary value problem with non-local condition for the second kind degenerated parabolic-hyperbolic equation, was investigated. The solution of the considered problem was searched in the form of solution of the first boundary value problem in the parabolic domain and also as the solution of modified Cauchy's problem in the hyperbolic domain. The special class of regular solutions of modified Cauchy's problem for hyperbolic type equation was introduced and the view of the solution simplified. Considered problem equivalently reduced (in the meaning uniqueness and existence of the solution) to Volterra's integral equation of the second kind. Necessary conditions were found for providing continuity of the kernel and the independent term of the obtained integral equation. The identities involving hyperheometric functions were proved by using Laplace transform. In the work, the properties of Bessel functions, hyperheometric functions, Euler's gamma function and Pohgammer's symbol were used. [ABSTRACT FROM AUTHOR]

In this paper, we study a non-local initial boundary value problem for a one-dimensional pseudohyperbolic equation. We first establish the existence and uniqueness of strong solution, then a numerical solutions for the system will be derived by using the finite-difference method. [ABSTRACT FROM AUTHOR]

Copyright of Cybernetics & Systems Analysis / Kibernetiki i Sistemnyj Analiz is the property of V.M. Glushkov Institute of Cybernetics of NAS of Ukraine and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The existence and uniqueness of regular solutions of non-local boundary value problems for the third order pseudoparabolic equations with variable coefficients are proved using the Riemann function method.

In this work, we study a boundary problem with non-local conditions, by relating values of the unknown function with various characteristics. The parabolic-hyperbolic equation with three lines of type changing is equivalently reduced to a system of Volterra integral equations of the second kind.

Some general conditions sufficient for unique solvability of the boundary-value problem for a system of linear functional differential equations of the second order are established. The class of equations considered covers, in particular, linear equations with transformed argument, integro-differential equations and neutral equations. An example is presented to illustrate the general theory.

We investigate the unique solvability of a class of nonlinear nonlocal differential equations associated with degenerate linear operator at the fractional Caputo derivative. For the main results, we use the theory of fractional calculus and (L, p)-boundedness technique that based on the analysis of both strongly (L, p)-sectorial operators and strongly (L, p)-radial operators. The obtained results are applicable to degenerate fractional Cauchy and Showalter-Sidorov problems in Banach spaces. Finally, we give an application described by time-fractional order Oskolkov system. [ABSTRACT FROM AUTHOR]

This research work is devoted to investigations of the existence and uniqueness of the solution of a non-local boundary value problem with discontinuous matching condition for the loaded equation. Considering parabolic-hyperbolic type equations involves the Caputo fractional derivative and loaded part joins in Riemann-Liouville integrals. The uniqueness of a solution is proved by the method of integral energy and the existence is proved by the method of integral equations. [ABSTRACT FROM AUTHOR]

This manuscript deals with a new set of sufficient conditions for the existence of solutions for a class of impulsive fractional neutral stochastic integro-differential systems (IFNSIDS) with nonlocal conditions (NLCs) and state-dependent delay (SDD) in Hilbert spaces. An example is provided to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

In this paper we consider the second order parabolic partial differential equation with constant coefficients subject to homogeneous Dirichlet boundary conditions and initial condition containing nonlocal integral term. We derive first and second order finite difference schemes for the parabolic problem, combining implicit and Crank-Nicolson methods with two discretizations of the integral term. One numerical example is presented to test and illustrate the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with existence of mild solutions for a first order impulsive neutral fractional integrodifferential inclusions with nonlocal initial conditions in α- norm. We assume that the linear part generates an analytic semigroup and transform it into an integral inclusion. By using a fixed point theorem for multi-valued maps due to Dhage, a main existence theorem is established. Finally, we present an example to illustrate this main theorem. [ABSTRACT FROM AUTHOR]

General conditions are obtained for the unique solvability of a non‐local boundary value problem for systems of linear functional differential equations. First Published Online: 14 Oct 2010

In this work, we study a boundary problem with non-local conditions, by relating values of the unknown function with various characteristics. The parabolic-hyperbolic equation with three lines of type changing is equivalently reduced to a system of Volterra integral equations of the second kind. [ABSTRACT FROM AUTHOR]

Abstract: The paper gives a new and natural method for the existence of multiple positive solutions for first order differential systems with non-local initial value conditions involving linear functionals. The case of higher order differential equations is also considered. The results are accompanied by numerical examples confirming the theory and proving for practice the importance of the bounds in solution localization. [Copyright &y& Elsevier]

General conditions are obtained for the unique solvability of a non-local boundary value problem for systems of linear functional differential equations. [ABSTRACT FROM AUTHOR]

Abstract: A numerical method based on an integro-differential equation and local interpolating functions is proposed for solving the one-dimensional wave equation subject to a non-local conservation condition and suitably prescribed initial-boundary conditions. To assess its validity and accuracy, the method is applied to solve several test problems. [Copyright &y& Elsevier]

A time-stepping dual-reciprocity boundary element method is proposed for solving approximately an inverse problem of determining an unknown control parameter p(t) which is the coefficient of the solution u(x,y,t) in a diffusion equation in a two-dimensional region R . The diffusion equation is to be solved subject to suitably prescribed initial-boundary conditions. The presence of the unknown coefficient p(t) requires an extra condition. This takes the form of a non-local condition expressed in terms of an integral over R . The method is applied to solve a specific test problem. [ABSTRACT FROM AUTHOR]