Integral Boundary Conditions Research Articles

Analysis of JS-contractions with applications to fractional boundary value problems

In this article, we modify JS-contractions by weakening the conditions on the function θ, where theta : ( 0,infty ) rightarrow ( 1,infty ) is a strictly increasing function. We prove fixed-point results for obtained contractions. Some examples are given to validate the results and modifications. We use our main theorem to establish the existence results for the solutions of the Atangana–Baleanu–Caputo fractional boundary value problem with integral boundary conditions. We also present a new definition of θ-Ulam stability and find the stability of our fractional boundary value problem.

A Non-Local Non-Homogeneous Fractional Timoshenko System with Frictional and Viscoelastic Damping Terms

We are devoted to the study of a non-local non-homogeneous time fractional Timoshenko system with frictional and viscoelastic damping terms. We are concerned with the well-posedness of the given problem. The approach relies on some functional analysis tools, operator theory, a priori estimates and density arguments. This work can be considered as a contribution to the development of energy inequality methods, the so-called a priori estimate method inspired from functional analyses and used to prove the well-posedness of mixed problems with integral boundary conditions.

On the existence of a positive solution to a boundary value problem for one nonlinear functional-differential equation of the second order

This article considers a boundary value problem for one non-linear second-order functional differential equation on the segment [0, 1] with an integral boundary condition at one of the ends of the segment. Using the well-known Go-Krasnoselsky theorem, sufficient conditions for the existence of at least one positive solution to the problem under consideration are established. A non-trivial example is given, illustrating the fulfillment of the conditions for the unique solvability of the problem posed.

Hybrid differential inclusion with nonlocal, infinite-point or Riemann-Stieltjes integral boundary conditions

Hybrid differential inclusion with nonlocal, infinite-point or Riemann-Stieltjes integral boundary conditions

APPLICATION OF CALCULATION METHODS FOR SOLVING APPLIED PROBLEMS OF HEAT AND MASS EXCHANGE IN COMPLEX SYSTEMS

The accuracy of the calculation and optimization of the objective function and its parameters when solving applied engineering problems depends on the accuracy of the formulation of the main optimization problem, calculation and applied optimization problems, as well as the accuracy of computational methods for their implementation. An increase in the considered features of applied optimization problems will complicate the setting and methods of implementing boundary value problems. Thus, for the implementation of modernized boundary value problems, it will be necessary to apply several computational methods that will create a computational structure. The main condition for constructing a physically based boundary value problem is to find and justify the conditions for the existence of a unique solution. To increase the efficiency of the use of methods of calculation and optimization of technical parameters, it is necessary to increase the number of considered features of calculation and applied optimization mathematical models for heat and mass transfer in technical systems. Along with the construction of boundary value problems, it is important to define and justify the conditions for the existence of a single solution.
 The research article deals with some aspects of solving applied problems of heat and mass transfer in technical systems. Nonlocal boundary value problems for inhomogeneous and homogeneous pseudodifferential equations in partial derivatives with integral boundary conditions are considered, methods of solving a nonlocal inhomogeneous boundary value problem are proposed, and the correctness conditions of this problem in the class of infinitely differentiable generalized functions of power growth are defined and proven. Proved conditions for the existence of a correct problem for pseudo-differential equations with an integral boundary condition. The research of this article should be applied for controlling possible risks when solving applied problems in technical systems, biotechnology and veterinary medicine.

On the Generalization of Tempered-Hilfer Fractional Calculus in the Space of Pettis-Integrable Functions

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology.

Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition

Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition

Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator

In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented in this study establish the existence, uniqueness, and Hyers–Ulam (HU) stability of the solution to the proposed equation. We achieved our main result by using the Banach contraction mapping principle and Krasonoselskii’s fixed point theorem. Furthermore, we introduce an application to demonstrate the validity of the results of our findings.

A non-uniform Haar wavelet method for a singularly perturbed convection–diffusion type problem with integral boundary condition on an exponentially graded mesh

A non-uniform Haar wavelet method for a singularly perturbed convection–diffusion type problem with integral boundary condition on an exponentially graded mesh

Solving 2D-integro-differential problems with nonlocal boundary conditions via a matrix formulated approach

A new operational matrix based approach is studied for numerical solution of 2D-integro-differential equations with non-local (integral) boundary conditions whose arise in some physical problems. Some important theoretical results are presented to reduce complexity and computational costs of the proposed method. We also give an error estimation which will be useful in estimating the error of approximate solution for the problems that we do not have any information about their exact solution. Illustrative numerical examples are also given to clarify the performance and accuracy of the new method.

Successive approximations and interval halving for fractional BVPs with integral boundary conditions

We study a system of non-linear fractional differential equations, subject to integral boundary conditions. We use a parametrization technique and a dichotomy-type approach to reduce the original problem to two “model-type” fractional boundary value problems with linear two-point boundary conditions. A numerical-analytic technique is applied to analytically construct approximate solutions to the “model-type” problems. The behaviour of these approximate solutions is governed by a set of parameters, whose values are obtained by numerically solving a system of algebraic equations. The obtained results are confirmed by an example of the fractional order problem that in the case of the second order differential equation models the Antarctic Circumpolar Current.

Coefficient inverse problem for the strongly degenerate parabolic equation

The coefficient inverse problem for the degenerate parabolic equation is investigated. The minor coefficient of this equation is the polynomial of the first power with respect to the space variable with two unknown time-dependent functions. The investigation is carried out under given inhomogeneous initial condition, Dirichlet boundary conditions and integral overdetermination conditions. We establish the conditions of the unique solvability to the named problem for the case of strong degeneration.

Fitted finite difference technique for singularly perturbed reaction diffusion equations of third order with integral boundary condition

AbstractWe have considered a third order singularly perturbed reaction diffusion equations with integral boundary condition. On a Shishkin mesh, a numerical technique based on a finite difference scheme is provided. The proposed approach is almost first order convergent. In the discrete norm, an error estimate is obtained. Numerical examples are provided to validate the theoretical estimates.

Vector nonlinear Schrödinger equation with an integrable defect and new integrable boundary conditions

The main purpose of the present paper is to study integrable defects and new integrable boundaries associated with the vector nonlinear Schrödinger (NLS) equation. For the first issue, we present a type I defect condition for the vector NLS equation by using a Bäcklund transformation (BT) frozen at defect location. Based on the canonical property of the BT, the integrability of our defect model is proved by virtue of the classical r-matrix method. The generating function of the infinite set of conserved quantities to the defect system is also constructed. For the second issue, we present several new integrable boundary conditions associated with the K(λ) matrices that are characterized by the presence of the vector NLS fields. These boundary conditions are constructed by imposing U(N) symmetry reduction on our defect condition and by dressing a boundary with our defect condition. The integrability of our new boundary conditions is proved by virtue of the Sklyanin’s approach. The Hamiltonian description of the vector NLS equation in the presence of our new boundary conditions is also discussed.

Existence of Solutions for Two-Point Integral Boundary Value Problems with Impulses

In this paper, we investigate the existence of at least one solution and at least two nonnegative solutions of impulsive differential equations with the two-point integral boundary conditions. We employ the recent fixed point theorems for the sum of two operators on Banach spaces. The applicability of the results is illustrated through an example.

Investigation of hybrid fractionalq-integro-difference equations supplemented with nonlocalq-integral boundary conditions

AbstractIn this article, we introduce and study a new class of hybrid fractionalqq-integro-difference equations involving Riemann-Liouvilleqq-derivatives, supplemented with nonlocal boundary conditions containing Riemann-Liouvilleqq-integrals of different orders. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present the existing criteria for solutions to the problem at hand by applying Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results. Some new results follow as special cases of this work.

On a Mixed Nonlinear Fractional Boundary Value Problem with a New Class of Closed Integral Boundary Conditions

On a Mixed Nonlinear Fractional Boundary Value Problem with a New Class of Closed Integral Boundary Conditions

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

In this work, we consider first-order singularly perturbed quasilinear Fredholm integro-differential equation with integral boundary condition. Building a numerical strategy with uniform ε-parameter convergence is our goal. With the use of exponential basis functions, quadrature interpolation rules and the method of integral identities, a fitted difference scheme is constructed and examined. The weight and remainder term are both expressed in integral form. It is shown that the method exhibits uniform first-order convergence of the perturbation parameter. Error estimates for the approximation solution are established and a numerical example is given to validate the theoretical findings.

System of fractional boundary value problems at resonance

In this paper, we investigate the existence of solutions for a system of fractional differential equations at resonance set on the interval [0, 1]. We associate a Dirichlet condition at $$t=0$$ and an integral boundary condition at $$t=1$$ . Our main existence theorem relies on the coincidence degree theory. A detailed study of the linear operators involved in the fixed point formulation is carried out. Finally, an example is included to illustrate the applicability of theoretical result.

Applicability of Darbo’s Fixed Point Theorem on the Existence of a Solution to Fractional Differential Equations of Sequential Type

In this article, we study the existence of the solution for the fractional differential equations of sequential type with nonlocal integral boundary conditions. The main results are established with the aid of Darbo’s fixed point theorem and Hausdorff’s measure of noncompactness method. The stability of the proposed fractional differential equation is also investigated via the Ulam–Hyer technique. In addition, an applied example that supports the theoretical results reached through this study is included.