Regular Boundary Value Problems Research Articles

The overdetermined Cauchy problem for the hyperbolic Gellerstedt equation

Abstract Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.

Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem

Abstract In this article, we study a class of variational inequality problems with non-Newtonian polytropic parabolic operators. We introduce a mapping with an adjustable parameter to control the polytropic term, which exactly meets the conditions of Leray-Schauder fixed point theory. At the same time, we construct a penalty function to transform the variational inequality into a regular parabolic initial boundary value problem. Thus, the existence is treated with a Leray-Schauder fixed point theory as well as a suitable version of Aubin-Lions lemma. Then, the uniqueness and stability of the solution are analyzed.

A hybrid reptile search algorithm and Levenberg–Marquardt algorithm based Haar wavelets to solve regular and singular boundary value problems

A hybrid reptile search algorithm and Levenberg–Marquardt algorithm based Haar wavelets to solve regular and singular boundary value problems

A new insight to the Hamiltonian systems with a finite number of spectral parameters

In this article, we introduce a new first-order differential equation containing a finite number of spectral parameters and some results on the solutions of this equation. In particular, with the aid of the nested-circles approach we share a lower bound for the number of linearly independent square-integrable solutions of the equation. We share some limit-point criterias. Moreover, we show that some known and unknown scalar and matrix differential equations can be embedded into this new first-order equation. Using the obtained results we present some additional results for some system of scalar multiparameter differential equations. Finally, we share some relations between the characteristic function of a regular boundary-value problem and the kernel of related integral operator.

Inverse spectral problems for functional-differential operators with involution

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems.

Eigenvalues of discontinuous third-order boundary value problems with eigenparameter-dependent boundary conditions

Regular third-order boundary value problems with eigenparameter-dependent boundary conditions and transmission conditions are investigated. It is obtained that the eigenvalues of the considered problems depend not only continuously but also smoothly on the parameters of the problems, i.e., the coefficients, the boundary conditions and transmission conditions. Moreover, the corresponding differential expressions for each parameters are given.

General characteristics of a fractal Sturm–Liouville problem

In this paper, we consider a regular fractal Sturm-Liouville boundary value problem. We prove the self-adjointness of the differential operator which is generated by the $F^\alpha$-derivative introduced in [32]. We obtained the $F^\alpha$-analogue of Liouville's theorem, and we show some properties of eigenvalues and eigenfunctions. We present examples to demonstrate the efficiency and applicability of the obtained results. The findings of this paper can be regarded as a contribution to an emerging field.

A hybrid augmented compact finite volume method for the Thomas–Fermi equation

A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm.

An effective scheme for solving a class of nonlinear doubly singular boundary value problems through quasilinearization approach

In this paper, an effective analytical method is introduced for solving a class of two-point nonlinear doubly singular boundary value problems (SBVPs) arising in various physical models. The method is the combination of Newton’s quasilinearization and Picard iteration method in which quasilinearization is utilized to reduce the nonlinear doubly SBVPs to a sequence of linear problems and then the Picard iteration method is used to obtain the approximate analytical solutions of linearized equations arising from quasilinearization method. The convergence analysis of the method is also discussed. To demonstrate the efficiency of the method, we consider various numerical examples arising in physical models including real-life problems. The numerical simulations justify the superiority and high performance of the method and the obtained results are compared with some other existing schemes. The comparisons reveal the applicability and effectiveness of the present work. The method also works quite efficiently for other nonlinear boundary value problems of any order. The method is successfully applied on second and third–order regular boundary value problems and the results are compared with the existing methods.

Numerical solution of doubly singular boundary value problems by finite difference method

In this paper, we propose a computational technique based on a combination of optimal homotopy analysis method (OHAM) and an iterative finite difference method (FDM) for a class of derivative-dependent doubly singular boundary value problems: $$\begin{aligned} (p(x)y^{\prime })^{\prime }= & {} q(x)f(x,y(x),y^{\prime }(x)),\quad 0\le x\le 1, \\ y^{\prime }(0)= & {} 0,~~ {\alpha }y(1)+{\beta }y^{\prime }(1)=B \end{aligned}$$ or $$\begin{aligned} y(0)=A,~~{\alpha }y(1)+{\beta }y^{\prime }(1)=B. \end{aligned}$$ The principal idea of this approach is to decompose the domain of the problem $$D=[0,1]$$ into two subdomains as $$D=D_{1}\cup D_{2}=[0,\gamma ]\cup [\gamma ,1]$$ ( $$\gamma $$ is the vicinity of the singularity). In the first domain $$D_{1},$$ we use OHAM to overcome the singularity behaviour at $$x=0$$ . In the second domain $$D_{2}$$ , a FDM is designed for solving the resulting regular boundary value problem. Convergence analysis of the method is carried out. Three nonlinear examples are considered to demonstrate the performance and accuracy of the proposed method. It is shown that the computational order of convergence of the FDM is two.

Variational regularization of damage models based on the emulated RVE

Material models exhibiting softening effects due to damage or localization share the problem of leading to ill-posed boundary value problems that lead to physically meaningless, mesh-dependent finite element results. It is thus necessary to apply regularization techniques that couple local behavior, described, e.g., by internal variables, at a spatial level. The common way to do this is to take into account higher gradients of the field variables, thus introducing an internal length scale. In this paper, we suggest a different approach to regularization that does not make use of any nonlocal enhancement like the inclusion of higher gradients or integration over local sub-domains nor of any classical viscous effects. Instead we perform an appropriate relaxation of the (condensed) free energy in a time-incremental setting which leads to a modified energy that is coercive and satisfies quasiconvexity in an approximate way. Thus, in every time increment a regular boundary value problem is solved. The proposed approach holds the same advantage as other methods, but with less numerical effort. We start with the theoretical derivation, discuss a rate-independent version of the proposed model and present details of the numerical treatment. Finally, we give finite element results that demonstrate the efficiency of this new approach.

Regular Boundary Value Problems for the Dirac Operator

Spectral problems for the Dirac operator specified on a finite interval with regular, but not strongly regular boundary conditions and a complex-valued integrable potential are studied. This work is aimed at finding the conditions under which the root function system forms a common Riesz basis rather than a Riesz basis with parentheses.

Basis properties of root functions of a regular fourth order boundary value problem

In this paper, we consider the following boundary value problem\[ y^{(4)}+q(x) y=\lambda y,~\ \ \ 0<x<1, \]\[ y^{\prime\prime\prime}\left(1\right)-\left(-1\right)^{\sigma}y^{\prime\prime\prime}\left(0\right)+\alpha y\left(0\right) =0, \]\[ y^{(s)}(1) -( -1) ^{\sigma}y^{(s) }( 0) =0,\ \ \ s=\overline{0,2}, \]where $\lambda $ is a spectral parameter, $q( x)\in L_{1}(0,1)$ is complex-valued function and $\sigma =0,1$. The boundary conditions of this problem are regular but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. When $\alpha\ne 0$, we proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a Riesz basis in the space $L_{2}( 0,1)$. Furthermore, we show that the system of root functions forms a basis in the space $L_{p}( 0,1)$, $1<p<\infty$ $(p\neq 2)$, under the conditions $\alpha\ne 0$ and $q( x) \in W_{1}^{1}( 0,1)$.

Highly Accurate Method for Solving Singular Boundary-Value Problems Via Padé Approximation and Two-Step Quartic B-Spline Collocation

We propose a highly accurate method to solve a class of singular two-point boundary-value problems having singular coefficients. These problems often arise when a partial differential equation is reduced to an ordinary differential equation by physical symmetry. The proposed method is based on Pade approximation and two-step collocation. The resolution domain is divided, Pade approximant is used on the subdomain in the vicinity of the singular point and provides a new boundary condition, and then, two-step quartic B-spline is used on the remaining subdomain where the problem is transformed to a regular boundary-value problem. The subdomains are chosen optimally using a suitable optimization procedure. Some examples are presented along with a comparison of the numerical results with other methods.

A new mixed MADM-Collocation approach for solving a class of Lane–Emden singular boundary value problems

In this paper, a new approach is proposed for solving a class of singular boundary value problems of Lane–Emden type. It is well known that the Adomian decomposition method (ADM) fails to provide a convergent series solution to strongly nonlinear boundary value problem in the wider region and the B-spline collocation method yields unsatisfactory approximation in the presence of singularity. To avoid these shortcomings of both methods, we propose a novel numerical method based on a combination of modified Adomian decomposition method and quintic B-spline collocation method to obtain more accurate solution of the problem under consideration. The principal idea of this approach is to decompose the domain of the problem $$D = [0, 1]$$ into two subdomains as $$D = D_{1} U D_{2}= [0, \delta ] U [\delta , 1]$$ ( $$\delta $$ is vicinity of the singularity). In the first domain $$D_1$$ , the underlying singular boundary value problem is efficiently tackled by a modified Adomian decomposition method. The intent is to apply the ADM in the smaller domain for finding a satisfactory solution. Finally, in the second domain $$D_2$$ , a collocation approach based on quintic B-spline basis function is designed for solving the resulting regular boundary value problem. The error estimation of the quintic B-spline interpolation is supplemented. In addition, six illustrative examples are presented to demonstrate the applicability and accuracy of the new method. It is shown that the resulting solutions appear to be higher accurate when compared to some existing numerical methods.

Conditions for the presence of negative eigenvalues in the Sturm-Liouville regular boundary value problem

Conditions for the presence of negative eigenvalues in the Sturm-Liouville regular boundary value problem

Extensions of a Minimal Third-Order Formally Symmetric Operator

In this paper, we consider some regular boundary value problems generated by a third-order differential equation and some boundary conditions. In particular, we construct maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal operator. Further using Lax–Phillips scattering theory and Sz.-Nagy–Foias characteristic function theory we prove a completeness theorem.

Regular third-order boundary value problems

In this paper, we consider some boundary value problems generated by third-order formally symmetric (self-adjoint) regular differential expression and separated, real-coupled and complex-coupled boundary conditions. It is shown that these problems generate self-adjoint operators. Moreover, the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced.

Regular Fractional Differential Equations in the Sobolev Space

Abstract In this paper regular fractional Sturm-Liouville boundary value problems are considered. In particular, new inner products are described in the Sobolev space and a symmetric operator is established in this space.

Constructing a basis from systems of eigenfunctions of one not strengthened regular boundary value problem

We investigate a nonlocal boundary value spectral problem for an ordinary differential equation in an interval. Such problems arise in solving the nonlocal boundary value problem for partial equations by the Fourier method of variable separation. For example, they arise in solving nonstationary problems of diffusion with boundary conditions of Samarskii-Ionkin type. Or they arise in solving problems with stationary diffusion with opposite flows on a part of the interval. The boundary conditions of this problem are regular but not strengthened regular. The principal difference of this problem is: the system of eigenfunctions is comlplete but not forming a basis. Therefore the direct applying of the Fourier method is impossible. Based on these eigenfunctions there is constructed a special system of functions that already forms the basis. However the obtained system is not already the system of the eigenfunctions of the problem. We demonstrate how this new system of functions can be used for solving a nonlocal boundary value problem on the example of the Laplace equation.