Class Of Boundary Value Problems Research Articles

Formal Theory of Cornered Asymptotically Hyperbolic Einstein Metrics

This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the usual infinite boundary and a corner where the two meet. On the finite boundary, a constant mean curvature umbilic condition is imposed. First, recent work of Nozaki, Takayanagi, and Ugajin is generalized and extended showing that such metrics cannot have smooth compactifications for generic corners embedded in the infinite boundary. A model linear problem is then studied: a formal expansion at the corner is derived for eigenfunctions of the scalar Laplacian subject to certain boundary conditions. In doing so, scalar ODEs are studied that are of relevance for a broader class of boundary value problems and also for the Einstein problem. Next, unique formal existence at the corner, up to order at least equal to the boundary dimension, of Einstein metrics in a cornered asymptotically hyperbolic normal form which are polyhomogeneous in polar coordinates is demonstrated for arbitrary smooth conformal infinity. Finally it is shown that, in the special case that the finite boundary is taken to be totally geodesic, there is an obstruction to existence beyond this order, which defines a conformal hypersurface invariant.

Existence of Solutions for Fractional Integro-Differential Equations with Non-Local Boundary Conditions

In this paper, we study the existence of solutions for a new class of boundary value problems of non-linear fractional integro-differential equations. The existence result is obtained with the aid of Schauder type fixed point theorem while the uniqueness of solution is established by means of contraction mapping principle. Then, we present some examples to illustrate our results.

Non-linear radiation effects on magnetic/non-magnetic nanoparticles with different base fluids over a flat plate

This work investigates the two-dimensional steady convective boundary layer flow and heat transfer of Newtonian/non-Newtonian base fluids with magnetic/non-magnetic nanoparticles over a flat plate which incorporates non-linear thermal radiation and slip effects. We considered magnetite and aluminium oxide as magnetic and non-magnetic nanoparticles suspending inside the two sorts of base fluids specifically Water and Sodium Alginate. For physical significance we analyzed the behavior on non-Newtonian profiles by employing Casson model individually. The particular intrigue lies in looking the impacts of non-linear thermal radiation on the behavior of the flow. The solution of wide class of boundary value problems are facilitated by the change of the partial differential equations administering the flow utilizing similarity transformations into ordinary differential equations. The ODE’s are numerically handled by applying fourth order Runge-Kutta integration scheme in association with shooting procedure. The novel results for the dimensionless velocity and temperature inside the boundary layer are exhibited graphically for various parameters that describe the flow. A graphical demonstration is given for the skin friction coefficient and the local Nusselt number.

Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line

This article investigates a new class of boundary value problems of one-dimensional lower-order nonlinear Hadamard fractional differential equations and nonlocal multi-point discrete and Hadamard integral boundary conditions. By using monotone iterative method, we not only seek the twin positive solutions of the problem but also show that the monotone iterative schemes converge to a unique positive solution of the problem. An error estimate formula is also given. For the illustration of the main results, we construct an example.

L<sup>1</sup>-Solutions of Boundary Value Problems for Implicit Fractional Order Differential Equations with Integral Conditions

The aim of this paper is to present new results on the existence of solutions for a class of boundary value problem for fractional order implicit di erential equations with integral conditions involving the Caputo fractional derivative. Our results are based on Schauder's xed point theorem and the Banach contraction principle fi xed point theorem.

Axisymmetric equilibrium of an isotropic elastic solid circular finite cylinder

Axisymmetric equilibrium of an elastic solid circular finite cylinder is one of the oldest problems in the theory of linear elasticity. Crosswise superposition is a well-known method that is used to solve this boundary value problem (BVP); however, technical realization of its underlying ideas still seems to be vague and the solution obtained by this method suffers from some convergence issues. In this study, we follow two main objectives via analyzing a benchmark problem where an isotropic elastic solid circular cylinder of finite length is subjected to normal lateral loading. The first goal is to add more insight into the method of crosswise superposition by extending the ideas that are used for solving classical BVPs via the superposition principle. For this purpose, a new unified approach that naturally gives rise to the subtle concept of corner conditions (CCs) in the context of crosswise superposition method is introduced. Another goal is to demonstrate the influence of CCs on the convergence of the solution obtained by the method of crosswise superposition. In this avenue, the Love function approach is used to convert the Navier equations for an isotropic elastic material to a single axisymmetric biharmonic equation. Next, a general solution for the axisymmetric biharmonic equation consisting of separable and non-separable solutions is presented in cylindrical coordinates. These two classes of solutions are used to construct the Love function and associated elastic fields through the unified approach. Numerical results reveal that considering the CCs can significantly affect the convergence rate of the solution on the boundaries of the cylinder. Furthermore, it is observed that the solution does not converge to the boundary data at the rims without considering the CCs. Far enough from the boundaries of the cylinder, the solution does not seem to be much different with or without taking the CCs into account.

Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter

In this paper, we investigate a class of boundary value problem of nonlinear Hadamard fractional differential equations with a parameter. By means of the properties of the Green function and Guo–Krasnosel’skii fixed-point theorem on cones, the existence and nonexistence of positive solutions are obtained. Finally, some examples are presented to show the effectiveness of our main results.

Spectral Invariance of Pseudodifferential Boundary Value Problems on Manifolds with Conical Singularities

We prove the spectral invariance of the algebra of classical pseudodifferential boundary value problems on manifolds with conical singularities in the $$L_{p}$$ -setting. As a consequence we also obtain the spectral invariance of the classical Boutet de Monvel algebra of zero order operators with parameters. In order to establish these results, we show the equivalence of Fredholm property and ellipticity for both cases.

A novel Ishikawa–Green’s fixed point scheme for the solution of BVPs

In this article, a new numerical approach is introduced for the numerical solution of a wide class of boundary value problems (BVPs). The underlying strategy of the algorithm is based on embedding an integral operator, defined in terms of Green’s function, into Ishikawa fixed point iteration scheme. The validity of the method is demonstrated by a number of examples that confirm the applicability and high efficiency of the method. The absolute error or residual error computations show that the current technique provides highly accurate approximations.

Solution of Boundary Value Problems in Cylinders with Two-Layer Film Inclusions

We consider the class of boundary value problems for elliptic, parabolic, and hyperbolic equations in cylinders separated by a two-layer film into two half-cylinders. We prove the existence and uniqueness theorem and express the solutions in terms of solutions to analogous classical problems in cylinders without films.

EXISTENCE RESULTS TO A CLASS OF HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS

This article is devoted to the study of existence results to a class of boundary value problems for hybrid fractional differential equations. A couple of hybrid fixed point theorems for the sum of three operators are used for proving the main results. Examples illustrating the results are also presented.

Numerical method to a class of boundary value problems

A class of boundary value problems can be transformed uniformly to a least square problem with Toeplitz constraint. Conjugate gradient least square, a matrix iteration method, is adopted to solve this problem, and the solution process is elucidated step by step so that the example can be used as a paradigm for other applications.

Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional q-Difference Equation

A class of boundary value problems of Caputo fractional q-difference equation is introduced. Green’s function and its properties for this problem are deduced. By applying these properties and the Leggett-Williams fixed-point theorem, existence criteria of three positive solutions are obtained. At last, some examples are given to illustrate the validity of our main results.

The Existence of Solutions for Boundary Value Problem of Fractional Functional Differential Equations with Delay

A class of boundary value problem for fractional functional differential equation with delay $ \left\{ {\begin{array}{*{20}c} {^{C} D^{\sigma } \omega (t) = f(t,\omega _{t} ),t \in [0,\zeta ],} \\ {\omega (0) = 0,\,\omega ^{\prime}(0) = 0,\,\omega ^{\prime\prime}(\zeta ) = 1,} \\ \end{array} } \right. $ is studied, where $ 2 < \sigma \le 3,\,\,^{c} D^{\sigma } $ devote standard Caputo fractional derivative. In this article, three new criteria on existence and uniqueness of solution are obtained by Banach contraction mapping principle, Schauder fixed point theorem and nonlinear alternative theorem.

Existence of nonnegative solutions for a nonlinear fractional boundary value problem

Abstract This paper deals with the existence of solutions for a class of boundary value problem (BVP) of fractional differential equation with three point conditions via Leray-Schauder nonlinear alternative. Moreover, the existence of nonnegative solutions is discussed.

Linear BVPs and SIEs for Generalized Regular Functions in Clifford Analysis

We study some properties of a regular function in Clifford analysis and generalize Liouville theorem and Plemelj formula with values in Clifford algebra An(R). By means of the classical Riemann boundary value problem and of the theory of a regular function, we discuss some boundary value problems and singular integral equations in Clifford analysis and obtain the explicit solutions and the conditions of solvability. Thus, the results in this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.

ЕВКЛИДОВО РАССТОЯНИЕ ДО ЗАМКНУТОГО МНОЖЕСТВА КАК МИНИМАКСНОЕ РЕШЕНИЕ ЗАДАЧИ ДИРИХЛЕ ДЛЯ УРАВНЕНИЯ ГАМИЛЬТОНА-ЯКОБИ

A combined (jointing analytical methods and computational procedures) approach to the construction of solutions in a class of boundary-value problems for a Hamiltonian-type equation is proposed. In the class of problems under consideration, the minimax (generalized) solution coincides with the Euclidean distance to the boundary set. The properties of this function are studied depending on the geometry of the boundary set and the differential properties of its boundary. Methods are developed for detecting pseudo-vertices of a boundary set and for constructing singular solution sets with their help. The methods are based on the properties of local diffeomorphisms and use partial one-sided limits. The effectiveness of the research approaches developed is illustrated by the example of solving a planar timecontrol problem for the case of a nonconvex target set with boundary of variable smoothness.

Existence results of fractional differential equations with Riesz–Caputo derivative

This paper is concerned with a class of boundary value problems for fractional differential equations with the Riesz–Caputo derivative, which holds two-sided nonlocal effects. By means of a new fractional Gronwall inequalities and some fixed point theorems, we obtained some existence results of solutions. Three examples are given to illustrate the results.

Some classes of singular integral equations of convolution type in the class of exponentially increasing functions

In this article, we study some classes of singular integral equations of convolution type with Cauchy kernels in the class of exponentially increasing functions. Such equations are transformed into Riemann boundary value problems on either a straight line or two parallel straight lines by Fourier transformation. We propose one method different from the classical one for the study of such problems and obtain the general solutions and the conditions of solvability. Thus, the result in this paper improves the theory of integral equations and the classical boundary value problems for analytic functions.

Chebyshev Pseudo-Spectral Method for Bratu’s Problem

In this paper, the Chebyshev pseudo-spectral (CPS) method is proposed for solving a class of boundary value problems. Specially, we apply the CPS method for the Bratu’s problem and obtain an approximate solution for this problem. The obtained solution can approximate the exact solution very accurately. By providing some theorems, we survey the feasibility and convergence of approximate solutions. The approximate results are compared with many other available methods. By calculating absolute errors, we show the efficiency of the method with respect to the other methods.