Class Of Boundary Value Problems Research Articles

A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented.

ЗАДАЧА НЕЙМАНА ДЛЯ НЕЛОКАЛЬНОГО БИГАРМОНИЧЕСКОГО УРАВНЕНИЯ

The solvability conditions for a class of boundary value problems for a nonlocal biharmonic equation in the unit ball with the Neumann conditions on the boundary are studied. The nonlocality of the equation is generated by some orthogonal matrix. The presence and uniqueness of a solution to the proposed Neumann boundary condition is examined, and an integral representation of the solution to the Dirichlet problem in terms of the Green's function for the biharmonic equation in the unit ball is obtained. First, some auxiliary statements are established: the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given, the representation of the solution to the Dirichlet problem in terms of this Green's function is written, and the values of the integrals of the functions perturbed by the orthogonal matrix are found. Then a theorem for the solution to the auxiliary Dirichlet problem for a nonlocal biharmonic equation in the unit ball is proved. The solution to this problem is written using the Green's function of the Dirichlet problem for the regular biharmonic equation. An example of solving a simple problem for a nonlocal biharmonic equation is given. Next, we formulate a theorem on necessary and sufficient conditions for the solvability of the Neumann boundary condition for a nonlocal biharmonic equation. The main theorem is proved based on two lemmas, with the help of which it is possible to transform the solvability conditions of the Neumann boundary condition to a simpler form. The solution to the Neumann boundary condition is presented through the solution to the auxiliary Dirichlet problem.

Some fixed point results for $ \alpha $-admissible extended $ \mathcal{Z} $-contraction mappings in extended rectangular $ b $-metric spaces

<abstract><p>In this paper, we introduce $ \alpha $-admissible extended $ \mathcal{Z} $-contraction in the extended rectangular $ b $-metric spaces, then we provide some other conditions in Theorem 3.1, which are different from that in Chifu et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, and obtain the existence and uniqueness of fixed point in such spaces. Moreover, some examples are given to show the validity of our main theorems, and we give some corollaries related to our main results. As an application, we apply our main results to solve the existence of solutions for a class of boundary value problems of second order ordinary differential equations.</p></abstract>

On One Generalization of the Multipoint Nonlocal Contact Problem for Elliptic Equation in Rectangular Area

Nonlocal contact problem for two‐dimensional linear elliptic equations is stated and investigated. The method of separation of variables is used to find the solution of a stated problem in the case of Poisson’s equation. Then, the more general problem with nonlocal multipoint contact conditions for elliptic equation with variable coefficients is considered, and the iterative method to solve the problem numerically is constructed and investigated. The uniqueness and existence of the regular solution are proved. The iterative method allows reducing the solution of a nonlocal contact problem to the solution of a sequence of classical boundary value problems.

A NEW EFFICIENT METHOD FOR TWO CLASSES OF SINGULAR CONVOLUTION INTEGRAL EQUATIONS OF NON-NORMAL TYPE WITH CAUCHY KERNEL

In this article, our task is to study the existence and Noethericity of solution for two classes of singular convolution integral equations with Cauchy kernels in the non-normal type case. To obtain the conditions of solvability for such equations, we establish regularity theory of solvability. By means of the theory of Fourier analysis, we will transform the equations into boundary value problems for holomorphic functions. The holomorphic solutions and conditions of solvability are obtained by using the method of complex analysis in class {0}. Moreover, we also discuss the asymptotic property of solution near nodes. Therefore, our work generalizes and improves the theories of integral equations and the classical boundary value problems for holomorphic functions.

Hilfer-Hadamard Nonlocal Integro-Multipoint Fractional Boundary Value Problems

This paper is concerned with the existence and uniqueness of solutions for a new class of boundary value problems, consisting by Hilfer-Hadamard fractional differential equations, supplemented with nonlocal integro-multipoint boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

Size-independent strain gradient effective models based on homogenization methods: Applications to 3D composite materials, pantograph and thin walled lattices

The notion of strain gradients provides a reliable model for capturing size effects and localization phenomena. The difficulty in identifying corresponding constitutive parameters, on the other hand, restricts the practical applicability of such theory. In this work, we aim at developing homogenization based strain-gradient continuum models to compute the effective Cauchy and strain gradient moduli for a wide class of 3D architectured materials and composites prone to strain gradient effects. The present homogenization method is based on a variational principle in linear elasticity articulated with Hill's lemma, which is extended by considering the generalized kinematics within the framework of periodic homogenization. The microscopic displacement field is decomposed into a polynomial function of the introduced macroscopic kinematic variables and the other is periodic fluctuation. The fluctuating displacement is expressed versus the macroscopic kinematic variables solving sequential classical and higher order unit cell boundary value problems. The computed effective elastic moduli in the current computing framework do not depend on the unit cell size, and are therefore intrinsic to the microstructure within the structure. 3D applications for inclusion based composites, pantographic lattices and thin-walled lattice structures showing pronounced strain gradient effects are carried out in order to exemplify the proposed homogenization strategy.

BOUNDARY VALUE PROBLEM FOR NONLINEAR CAPUTO-HADAMARD FRACTIONAL DIFFERENTIAL EQUATION WITH HADAMARD FRACTIONAL INTEGRAL AND ANTI-PERIODIC CONDITIONS

The aim of this work is to study a class of boundary value problem including a fractional order differential equation involving the Caputo-Hadamard fractional derivative. Suffcient conditions will be presented to guarantee the existence and uniqueness of solution of this fractional boundary value problem. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem

In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q-differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution by means of the Schauder fixed point theorem. Then, the stability of solution will be discussed under the perturbations of boundary condition, a function existing in the problem, and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Wave motion of pre-stressed compressible hyperelastic cylindrical shell under radial perturbation

The radial propagation of the steady-state wave motion is studied for a pre-stressed cylindrical shell composed of the isotropic compressible neo-Hookean material. The radial static deformation of the cylindrical shell and radial wave motion of the pre-stressed cylindrical shell are investigated. The mathematical model is established as a class of boundary value problems of the second-order nonlinear ordinary differential equation (ODE). Based on the theory of the large elastic deformation superposed by the small harmonic deformation, a radial dynamic displacement is added to the pre-stressed cylindrical shell. The existence of the steady-state wave motion is conducted. Numerical solutions are obtained by using the multiple shooting method. In addition, the influences of the static load, material and structural parameters on the steady-state wave motion are discussed. Numerical results indicate that as the static load increases, the frequency of the steady-state wave motion increases and the amplitude decreases in the compressible hyperelastic cylindrical shell. The material parameter plays a significant role in the frequency and amplitude of the steady-state wave motion.

Implicit Hybrid Fractional Boundary Value Problem via Generalized Hilfer Derivative

This research paper is dedicated to the study of a class of boundary value problems for a nonlinear, implicit, hybrid, fractional, differential equation, supplemented with boundary conditions involving general fractional derivatives, known as the ϑ-Hilfer and ϑ-Riemann–Liouville fractional operators. The existence of solutions to the mentioned problem is obtained by some auxiliary conditions and applied Dhage’s fixed point theorem on Banach algebras. The considered problem covers some symmetry cases, with respect to a ϑ function. Moreover, we present a pertinent example to corroborate the reported results.

On a class of boundary value problems under ABC fractional derivative

In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results.

EXISTENCE THEORY TO A CLASS OF FRACTIONAL ORDER HYBRID DIFFERENTIAL EQUATIONS

In this paper, we develop the theory of fractional order hybrid differential equations involving Riemann–Liouville differential operators of order [Formula: see text]. We study the existence theory to a class of boundary value problems for fractional order hybrid differential equations. The sum of three operators is used to prove the key results for a couple of hybrid fixed point theorems. We obtain sufficient conditions for the existence and uniqueness of positive solutions. Moreover, examples are also presented to show the significance of the results.

Gauge freedom in magnetostatics and the effect on helicity in toroidal volumes

Magnetostatics defines a class of boundary value problems in which the topology of the domain plays a subtle role. For example, representability of a divergence-free field as the curl of a vector potential comes about because of homological considerations. With this in mind, we study gauge freedom in magnetostatics and its effect on the comparison between magnetic configurations through key quantities such as the magnetic helicity. For this, we apply the Hodge decomposition of k-forms on compact orientable Riemaniann manifolds with smooth boundary, as well as de Rham cohomology, to the representation of magnetic fields through potential one-forms in toroidal volumes. An advantage of the homological approach is the recovery of classical results without explicit coordinates and assumptions about the fields on the exterior of the domain. In particular, a detailed construction of minimal gauges and a formal proof of relative helicity formulas are presented.

A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions

This paper is a survey of the recent results of the author for various classes of boundary value problems for Hilfer fractional differential equations and inclusions of fractional order in (1,2] supplemented with different kinds of nonlocal boundary conditions.

A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH A DOMINANT MIXED DERIVATIVE

In this article, we consider the Goursat problem with nonlocal integral conditions for a hyperbolic equation with a dominant mixed derivative. Research methods of solvability of classical boundary value problems for partial differential equations cannot be applied without serious modifications. The choice of a research method of solvability of a nonlocal problem depends on the form of the integral condition. In the process of developing methods that are effective for nonlocal problems, integral conditions of various types were identified [1]. The solvability of the nonlocal Goursat problem with integral conditions of the first kind for a general equation with dominant mixed derivative of the second order was investigated in [2]. In our problem, the integral conditions are nonlocal conditions of the second kind, therefore, to investigate the solvability of the problem, we propose another method, which consists in reducing the stated nonlocal problem to the classical Goursat problem, but for a loaded equation. In this article, we obtain conditions that guarantee the existence of a unique solution of the problem. The main instrument of the proof is the a priori estimates obtained in the paper.

A unified approach to study the existence and numerical solution of functional differential equation

In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method. The approach used for the third order nonlinear functional differential equation can be applied to functional differential equations of any orders.

Existence of Positive Solutions for a Class of Boundary Value Problems with $$\boldsymbol{p}$$-Laplacian in Banach Spaces

Existence of Positive Solutions for a Class of Boundary Value Problems with $$\boldsymbol{p}$$-Laplacian in Banach Spaces

Boundary value problem for nonlinear fractional differential equations involving Erdélyi-Kober derivative on unbounded domain

In this paper, we establish sufficient conditions for the existence of bounded solution for a class of boundary value problem for nonlinear fractional differential equations involving the Erdélyi-Kober differential operator on unbounded domain. Our results are based on a fixed point theorem of Schauder combined with the diagonalization argument method in a special Banach space. To that end, an example is presented to illustrate the usefulness of our main results.

A new finite difference method for computing approximate solutions of boundary value problems including transition conditions

This article is aimed at computing numerical solutions of new type of boundary value problems (BVPs) for two-linked ordinary differential equations. The problem studied here differs from the classical BVPs such that it contains additional conditions at the point of interaction, so-called transition conditions. Naturally, such type of problems is much more complicated to solve than classical problems. It is not clear how to apply the classical numerical methods to such type of boundary value transition problems (BVTPs). Based on the finite difference method (FDM) we have developed a new numerical algorithm for computing numerical solution of BVTPs for two-linked ordinary differential equations. To demonstrate the reliability and efficiency of the presented algorithm we obtained numerical solution of one BVTP and the results are compared with the corresponding exact solution. The maximum absolute errors (MAEs) are presented in a table.