Types Of Boundary Value Problems Research Articles

A note on Kirchhoff type boundary value problem involving Riemann-Liouville fractional derivative

In this paper, we study some nonlinear Kirchhoff boundary value problems of fractional differential equations involving Riemann Liouville operator. Under appropriate assumptions on the functions in the given problem, we establish the existence of solutions using variational methods combined with the mountain pass theorem. Moreover, an illustrative example is presented to prove the validity of the main result.

Analytical solution for diffraction of magnetoelastic plane waves by a rigid strip in self-reinforced medium: a contour integration method

This study analytically addresses the diffraction of magnetoelastic plane waves by a rigid strip in a self-reinforced infinite elastic medium. Formulation of the studied model includes mixed type boundary value problem and has been solved by the method of integral equation. The expressions of vertical diffracted displacement and normal stress are established in the closed form by employing the contour integration technique. For the sake of validation, the deduced expressions of vertical diffracted displacement and normal stress are reduced to the special cases, such as without magnetoelastic coupling as well as anisotropy and matched with the pre-established results. The diffraction patterns for the vertical displacement component in the considered material medium are calculated numerically, and its dependence on various affecting parameters viz. magnetoelastic coupling parameter, phase velocity, wave number of the propagating magnetoelastic plane wave, and distance has been depicted graphically for self-reinforced and isotropic materials. Moreover, the influence of anisotropy of the infinite medium has been portrayed remarkably by the comparative study.

Asymptotic analysis of the narrow escape problem in general shaped domain with several absorbing necks

This paper considers the two-dimensional narrow escape problem in a domain which is composed of a relatively big head and several absorbing narrow necks. The narrow escape problem is to compute the mean first passage time (MFPT) of a Brownian particle traveling from inside the head to the end of the necks. The original model for MFPT is to solve a mixed Dirichlet–Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper, we compute the MFPT by solving an equivalent Neumann–Robin type boundary value problem. By solving the new model, we obtain the three-term asymptotic expansion of the MFPT. We also conduct numerical experiments to show the accuracy of the high order expansion. As far as we know, this is the first result on high order asymptotic solution for narrow escape problem (NEP) in a general shaped domain with several absorbing neck windows. This work is motivated by Li (2014 J. Phys. A: Math. Theor. 47 505202), where the Neumann–Robin model was proposed to solve the NEP in a domain with a single absorbing neck.

Boundary value problems of quaternion-valued differential equations: solvability and Green’s function

This paper is associated with Sturm–Liouville type boundary value problems and periodic boundary value problems for quaternion-valued differential equations (QDEs). Employing the theory of quaternionic matrices, we prove the conditions for the solvability of the linear boundary valued problem and find Green’s function.

Strain regularisation using a non-local method in Coupled Eulerian-Lagrangian analyses

Conventional finite element analyses generally suffer from mesh dependency when considering softening effects. Non-local strain regularisation techniques have been developed to address this issue, which are generally complex to implement. In view of this, this paper introduces an more efficient procedure for implementation of a non-local method in Abaqus for Coupled Eulerian-Lagrangian (CEL) large deformation finite element undrained analyses. Results from a series of biaxial compression simulations demonstrate that the non-local method with a strain-softening Tresca model in CEL avoids mesh dependency. Owing to the stationary Eulerian element and the built-in mapping algorithm in CEL, high computational efficiency is achieved, adding no more than 12% to the computational cost. Guidance is provided on selection of internal length scales, element sizes and the search radius to ensure efficient non-local calculations, and it is shown that a softening scaling rule can also be used to allow use of practical mesh densities for some boundary value problems. Simulations of a number of classical geotechnical problems demonstrate the type of boundary value problems where the non-local method can effectively mitigate the mesh dependency, whilst also highlighting that the method fails to control the strain localisation that develops at soil-structure interfaces due to geometrical issues.

Uniform approximations and effective boundary conditions for a high-contrast elastic interface

A difficulty in the theory of a thin elastic interface is that series expansions in its thickness become disordered in the high-contrast limit, i.e. when the interface is much softer or much stiffer than the media on either side. We provide a mathematical analysis of such series for an annular coating around a cylindrical fibre embedded in an elastic matrix subject to biaxial forcing. We determine the order of magnitude of successive terms in the series, and hence the terms that need to be retained in order to ensure that every neglected term is smaller in order of magnitude than at least one retained term. In this way, we obtain uniform approximations for quantities such as the jump in the displacement and stress across the coating, and explain some peculiarities that have been observed in numerical work. A key finding is that it is essential to distinguish three types of boundary-value problem, corresponding to ‘distant forcing’, ‘localized forcing’ and ‘the homogeneous problem’, since they give different patterns of disorder in the corresponding series expansions. This provides a meaningful correspondence between physical principles and our mathematical results.

FDM data driven U-Net as a 2D Laplace PINN solver

Efficient solution of partial differential equations (PDEs) of physical laws is of interest for manifold applications in computer science and image analysis. However, conventional domain discretization techniques for numerical solving PDEs such as Finite Difference (FDM), Finite Element (FEM) methods are unsuitable for real-time applications and are also quite laborious in adaptation to new applications, especially for non-experts in numerical mathematics and computational modeling. More recently, alternative approaches to solving PDEs using the so-called Physically Informed Neural Networks (PINNs) received increasing attention because of their straightforward application to new data and potentially more efficient performance. In this work, we present a novel data-driven approach to solve 2D Laplace PDE with arbitrary boundary conditions using deep learning models trained on a large set of reference FDM solutions. Our experimental results show that both forward and inverse 2D Laplace problems can efficiently be solved using the proposed PINN approach with nearly real-time performance and average accuracy of 94% for different types of boundary value problems compared to FDM. In summary, our deep learning based PINN PDE solver provides an efficient tool with various applications in image analysis and computational simulation of image-based physical boundary value problems.

Differential-difference systems in the analysis of weak solvability of initial-boundary value problems with a spatial variable in a network-like domain

In the work, the approach and the corresponding methods, which make it possible to construct a priori estimates of weak solutions of a differential-difference system with a spatial variable varying in a multidimensional network-like domain are indicated. Such estimates in spaces of summable functions are used to find solvability conditions for boundary value problems of various types for differential-difference systems. In addition, a priori estimates are used to justify the application of the method of discretization with respect to the time variable (semi-discretization) to the analysis of the weak solvability of initial-boundary value problems and the subsequent construction of approximations of weak solutions. The rationale for the approach used is the fact that in a fairly wide class applied analysis of the problems of transporting continuous media networks-like carriers, the representation of mathematical models of the process using the formalisms of differential-difference systems is the only tool for effectively solving these problems. For example, the reduction of a differential system (initial-boundary value problem) to the corresponding differential-difference system makes it possible not only to significantly simplify the analysis of problems of optimal control of a differential system (since this analysis reduces to studying the problem of optimal control of a system of elliptic equations), but also, using classical methods of control theory for elliptic systems, algorithmize the original problem. The reduction used often facilitates establishing the conditions for the existence and uniqueness of optimal control of a differential system. These problems also include a fairly large range of studies of non-stationary network-like hydrodynamic processes and flow phenomena. As an illustration of the approach used and the results obtained, the analysis of the solvability of the linearized Navier–Stokes system is given and the ways of studying the nonlinear Navier–Stokes system are indicated.

Generalized differential transformation method for solving two-interval Weber equation subject to transmission conditions

The main goal of this study is to adapt the classical differential transformation method to solve new types of boundary value problems. The advantage of this method lies in its simplicity, since there is no need for discretization, perturbation or linearization of the differential equation being solved. It is an efficient technique for obtaining series solution for both linear and nonlinear differential equations and differs from the classical Taylor’s series method, which requires the calculation of the values of higher derivatives of given function. It is known that the differential transformation method is designed for solving single interval problems and it is not clear how to apply it to many-interval problems. In this paper we have adapted the classical differential transformation method for solving boundary value problems for two-interval differential equations. To substantiate the proposed new technique, a boundary value problem was solved for the Weber equation given on two non-intersecting segments with a common end, on which the left and right solutions were connected by two additional transmission conditions.

Lyapunov-type inequalities for $(\mathtt{n},\mathtt{p})$-type nonlinear fractional boundary value problems

This paper establishes Lyapunov-type inequalities for a family of two-point $(\mathtt{n},\mathtt{p})$-type boundary value problems for Riemann-Liouville fractional differential equations. To demonstrate how the findings can be applied, we provide a few examples, one of which is a fractional differential equation with delay.

FRACTIONAL DISSIPATIVE STURM-LIOUVILLE PROBLEMS WITH DISCONTINUITY AND EIGEN-DEPENDENT BOUNDARY CONDITIONS

This paper studies two main fractional discontinuous dissipative Sturm-Liouville type boundary-value problems with boundary conditions and transmission conditions. In both types of research, with the aid of the operator theory, we define different classes of new inner products by combining the parameters in the boundary and transmission conditions, then the boundary value problems are transferred to operators in the Hilbert spaces such that the eigenvalues and eigenfunctions of the main problem coincide with those of operators. And we prove those of operators are dissipative. Moreover, the fundamental solutions are constructed and the uniqueness of the solutions of the problem is also given.

Solving systems of coupled nonlinear Atangana–Baleanu-type fractional differential equations

In this work, we investigate two types of boundary value problems for a system of coupled Atangana–Baleanu-type fractional differential equations with nonlocal boundary conditions. The fractional derivatives are applied to serve as a nonlocal and nonsingular kernel. The existence and uniqueness of solutions for proposed problems using Krasnoselskii’s and Banach’s fixed-point approaches are established. Moreover, nonlinear analysis is used to build the Ulam–Hyers stability theory. Subsequently, we discuss two compelling examples to demonstrate the utility of our study.

Short Proofs of Explicit Formulas to Boundary Value Problems for Polyharmonic Equations Satisfying Lopatinskii Conditions

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B1 a Green function is constructed in the cases c>0, c∉−N, where c is the coefficient in front of u in the boundary condition ∂u∂n+cu=f. To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λu+cu=f. Elliptic boundary value problems for Δmu=0 in B1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δu=0, Δ2u=0 and Δ3u=0 in B1 as well as some additional results from the theory of spherical functions are proposed.

Existence of vortices for Schrödinger equations with logarithmic and saturable nonlinearity

In this paper, we study the existence of stationary vortex wave solutions of two kinds of nonlinear Schrödinger equations. For the first one, which is equipped with logarithmic nonlinearity arising from Bose–Einstein condensation, we consider two types of boundary value problems. In both cases, we establish the existence of positive solutions through a direct minimization method. For the second one, with a saturable nonlinearity originating from geometric optics, we use a constrained minimization approach to establish the existence of vortex wave solutions. Moreover, some explicit estimates for the bound of the wave propagation constant are derived.

WITHDRAWN: Generalized Vieta-Fibonacci quasilinearization method for solutions of third-order singular Emden–Fowler type boundary value problems

In this article, we apply a generalized Vieta-Fibonacci (GVF) quasilinearization method to explore the numerical solution of third-order singular and nonlinear boundary value problems. Firstly, we use the quasilinearization technique to convert the governing nonlinear boundary value problem into a class of linearized equations. Then we build a collocation technique based on GVF functions to solve the resultant sequence of linearized equations. Convergence result of the method is established with respect to L 2 norm. Four test problems are considered to illustrate the applicability, efficiency and robustness of the proposed algorithm and to validate the theoretical estimate as well by comparing it with previous work.

Integral Form of the Heat Transfer Equation With Arbitrarily Moving Boundary and Arbitrary Heat Source

Abstract For the first time, an integral form of one-dimensional heat transfer equation in a semi-infinite domain with a boundary, moving arbitrarily in time, and a heat source, depending arbitrarily on time and space location, is obtained. The obtained integral equation relates time histories of the temperature and its gradient at the boundary of the domain with the temperature at any given point inside or at the boundary of the domain. In the latter case, it delivers closed form integral equation for the rate of boundary movement in nonlinear problems where the time history of boundary movement is one of problem unknowns. The obtained equation accounts explicitly for the presence of an arbitrary heat source in the domain, while other existing methods do not allow a closed integral formulation to be obtained in such a case. The equation may be used for an analytical investigation of several types of boundary value problems (BVPs), as well as for numerical solution of such problems. Particular cases of this equation with a trivial heat source are known to demonstrate chaotic behavior. It is expected that the same is true for some nontrivial heat source functions, and this conjecture will be explored in subsequent publications.

EXISTENCE AND UNIQUENESS SOLUTION FOR THREE-POINT HADAMARD-TYPE FRACTIONAL VOLTERRA BVP

In this paper we study the existence and uniqueness solution for a first kind fractional Volterra boundary value problem involving Hadamard type and three-point boundary conditions. Our analysis is based on Krasnoselskii-Zabreiko’s fixed point theorem and Banach contraction principle. As an application we discuss a Hadamard type boundary value problem with fractional integral boundary conditions. We emphasize that our results are new in the context of Hadamard fractional calculus and are well illustrated with the aid of examples.

A mathematical study of diffusive logistic equations with mixed type boundary conditions

<p style='text-indent:20px;'>The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.</p>

Boundary Value Problems of Thermoelasticity for Porous Sphere and for A Space with Spherical Cavity

This article is concerned with the coupled linear quasi-static theory of thermoelasticity for porous materials under local thermal equilibrium. The system of equations is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of governing equations is expressed in terms of displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. The present paper is devoted to construct explicit solutions of the quasi-static boundary value problems (BVPs) of coupled theory of thermoelasticity for a porous elastic sphere and for a space with a spherical cavity. In this research the regular solution of the system of equations for an isotropic porous material is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The basic boundary value problems (the Dirichlet type boundary value problem for a sphere and the Neumann type boundary value problem for a space with a spherical cavity) are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent series.

A comprehensive static modeling methodology via beam theory for compliant mechanisms

Compliant Mechanisms (CMs) present several desired properties for mechanical applications only depending on elastic deformation of the involved compliant beams/flexures. As reported in the current literature, most CM designs utilize straight beams and initially curved beams (ICBs) as the fundamental flexible members. In CM research community, many great contributions regarding modeling these elementary flexible members have been achieved. In this paper, a comprehensive modeling methodology, based on beam theory, has been established to characterize the static planar deflection of slender beam. Then such a methodology has been applied to solve 8 loading scenarios of large beam-deflection problems that exist in the design of CMs. Essentially speaking, all these beam-deflection problems are treated as a type of boundary value problems (BVPs) of an ordinary differential equation (ODE) and solved by a modified collocation method. After that, this methodology has been used to model some representative CMs with large-deflection strokes, such as compliant parallelograms.