Nonhomogeneous Boundary Value Problems Research Articles

LYAPUNOV-TYPE INEQUALITY FOR CERTAIN HALF-LINEAR LOCAL FRACTIONAL ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we establish a Lyapunov-type inequality for the half-linear local fractional ordinary differential equation based on the formulation of the local fractional derivative. In addition, we apply the inequality to investigate the non-existence and uniqueness of solutions for related homogeneous and non-homogeneous local fractional boundary value problems.

Non-homogeneous initial boundary value problems for the biharmonic Schrödinger equation on an interval

Non-homogeneous initial boundary value problems for the biharmonic Schrödinger equation on an interval

Critical Point Approaches for a Class of Differential Equations with Sturm-Liouville Type Nonhomogeneous Boundary Conditions

A class of p-Laplacian equations with Sturm-Liouville type nonhomogeneous boundary value problem with nonlinear derivative depending on two control parameters is investigated. Existence and multiplicity of solutions are discussed by means of variational methods and critical point theory. Two examples supporting our theoretical results are also presented.

Solution of infinite system of third-order three-point nonhomogeneous boundary value problem in weighted sequence space $${{\varvec{bv}}}_{{{\varvec{p}}}}^{\omega }$$

Solution of infinite system of third-order three-point nonhomogeneous boundary value problem in weighted sequence space $${{\varvec{bv}}}_{{{\varvec{p}}}}^{\omega }$$

An analytical method for nonlinear and nonhomogeneous boundary value problems of plates

An analytical method for nonlinear and nonhomogeneous boundary value problems of plates

Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval

Abstract We have established the existence and uniqueness of the local solution for (0.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 , \left\{\begin{array}{ll}{\partial }_{t}u+{\partial }_{x}^{5}u-u{\partial }_{x}u=0,& 0\lt x\lt 1,\hspace{1.0em}t\gt 0,\\ u\left(x,0)=\varphi \left(x),& 0\lt x\lt 1,\\ u\left(0,t)={h}_{1}\left(t),u\left(1,t)={h}_{2}\left(t),\hspace{0.33em}{\partial }_{x}u\left(1,t)={h}_{3}\left(t),& \\ {\partial }_{x}u\left(0,t)={h}_{4}\left(t),\hspace{0.33em}{\partial }_{x}^{2}u\left(1,t)={h}_{5}\left(t),& t\gt 0,\end{array}\right. in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: Can the local solution be extended to a global one? This article will address this question. First, through a series of logical deductions, a global a priori estimate is established, and then the local solution is naturally extended to a global solution.

On the characteristic functions and Dirchlet-integrable solutions of singular left-definite Hamiltonian systems

In this work, a singular left-definite Hamiltonian system is considered and the characteristic-matrix theory for this Hamiltonian system is constructed. Using the results of this theory we introduce a lower bound for the number of Dirichlet-integrable solutions. Moreover we share a relation between the kernel of the solution of the nonhomogeneous boundary value problem and the characteristic-matrix.

Initial-boundary value problem for the Hirota equation posed on a finite interval

In this paper, we turn our attention to a nonhomogeneous initial boundary value problem for Hirota equation posed on a bounded interval (0,L). In particular, the explicit solution formula of linear nonhomogeneous boundary value problem is established by Laplace transform. Using space L2(0,T;H0s−1(0,1)), which is preparing for trilinear estimates and Lions-Magenes interpolation theorem, we prove the local existence, uniqueness, and Lipschitz continuous in C(0,T;Hs(0,1))∩L2(0,T;Hs+1(0,1)) corresponding to the initial and boundary data. Moreover, the local solution extends to a global one by a priori bound.

Guaranteed Lower Eigenvalue Bounds for Steklov Operators Using Conforming Finite Element Methods

Abstract For the eigenvalue problem of the Steklov differential operator, an algorithm based on the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed lower eigenvalue bounds utilize the a priori error estimation for FEM solutions to non-homogeneous Neumann boundary value problems, which is obtained by constructing the hypercircle for the corresponding FEM spaces and boundary conditions. Numerical examples demonstrate the efficiency of our proposed method.

Study of non-Newtonian synovial fluid flow by a recursive approach

This study analyzes the non-Newtonian synovial fluid flow between the joints in a synovitis, which is a diseased condition due to inflammation of synovial membrane. It is assumed in this study that the secretion of synovial fluid through the inflamed synovial membrane is a linear function of the membrane length. The mathematical modeling of synovial fluid through a synovial membrane is made by the non-Newtonian Linear Phan-Thien–Tanner (LPTT) fluid model through a thin conduit having permeable walls. The nonlinear flow of LPTT fluid gives the non-homogeneous complex boundary value problem, and the recursive approach is used to solve the problem. The flow of synovial fluid along and across the membrane is calculated under the inflamed membrane, and results are displayed through graphs. The axial pressure required for the non-Newtonian fluid flow and deformation of synovial fluid that produces the shearing forces near the synovial membrane are also calculated. The purpose of this research is to observe the shear stress on the synovial fluid and inflammation rate on the flow along the membrane at different position and pressure required for the flow of synovial fluid in diseased condition. The mathematical and graphical results for pressure, flow, volume flux, and streamline are calculated and plotted using the software MATHEMATICA. This study is very helpful for the biomedical engineers to measure the compression force and shear stress on the synovial fluid in a diseased condition and can be controlled by the viscosity of the synovial fluid.

Non-homogeneous boundary value problems of the Kawahara equation posed on a finite interval

Considered in this paper is the initial boundary value problem (IBVP) of the Kawahara equation, a class of the fifth order KdV equation, posed on a finite interval, (0.1)ut+ux+βuxxx+uxxxxx+uux=0,0<x<L,t>0,u(x,0)=ϕ(x), subject to the non-homogeneous boundary conditions (0.2)Bju=hj(t),j=1,2,3,4,5t>0where Bju=∑k=04(ajk∂xku(0,t)+bjk∂xku(L,t)),j=1,2,3,4,5and ajk,bjk(k=0,1,2,3,4andj=1,2,3,4,5) are real constants. Under some general assumptions imposed on the coefficients ajk,bjk, the IBVP (0.1)–(0.2) is shown to be locally well posed in the space Hs(0,L) for any s≥0 with naturally compatible ϕ∈Hs(0,L) and boundary values hj,j=1,2,3,4,5 belonging to some appropriate spaces with optimal regularity. The sharp Kato smoothing properties (due to Kenig, Ponce and Vega (Kenig et al., 1991; Kenig et al., 1993)) of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line R, vt+βvxxx+vxxxxx=g(x,t),v(x,0)=ψ(x),x,t∈R,have played an important role in establishing the well-posedness of the IBVP (0.1)–(0.2).

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an L^infty -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in {mathbb {R}}^{ntimes n} with zero matrix traces. We analyze, in L^2-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in {mathbb {R}}^n, nge 2 (n=2,3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.

Multiplicity of solutions for a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the p ( x )-Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

Existence of Positive Solutions for a Singular Second-Order Changing-Sign Differential Equation on Time Scales

In this paper, we focus on the existence of positive solutions for a boundary value problem of the changing-sign differential equation on time scales. By constructing a translation transformation and combining with the properties of the solution of the nonhomogeneous boundary value problem, we transfer the changing-sign problem to a positone problem, then by means of the known fixed-point theorem, several sufficient conditions for the existence of positive solutions are established for the case in which the nonlinear term of the equation may change sign.

On the Existence of Helical Invariant Solutions to Steady Navier--Stokes Equations

In this paper, we investigate the nonhomogeneous boundary value problem for the steady Navier-Stokes equations in a helically symmetric spatial domain. When data is assumed to be helical invariant and satisfies the compatibility condition, we prove this problem has at least one helical invariant solution.

Solving the BVP to a class of second-order linear fuzzy differential equations under granular differentiability concept1

Based on the granular derivative and the horizontal membership function of fuzzy-number-valued-function, the existence and uniqueness of solutions of two-point boundary value problem (BVP) for a class of second-order linear fuzzy ordinary differential equations are given, including the homogeneous BVP, the semi-homogeneous BVP and the non-homogeneous BVP. However, this is somewhat different from ordinary differential equations. In fact, we can think of it simply as the transformation from a number to a set, or we can think of it macroscopically as the transformation from a point to a plane. At the same time, appropriate examples are given to illustrate this conclusion.

Lower regularity solutions of the non-homogeneous boundary-value problem for a higher order Boussinesq equation in a quarter plane

We continue to study the initial–boundary-value problem of the sixth order Boussinesq equation in a quarter plane with non-homogeneous boundary conditions: utt−uxx+βuxxxx−uxxxxxx+(u2)xx=0,x,t∈R+,u(x,0)=φ(x),ut(x,0)=ψ′′(x),u(0,t)=h1(t),uxx(0,t)=h2(t),uxxxx(0,t)=h3(t),where β=±1. We show that the problem is locally analytically well-posed in the space Hs(R+) for any s>−34 with the initial-value data (φ,ψ)∈Hs(R+)×Hs−1(R+)and the boundary-value data (h1,h2,h3)∈Hs+13(R+)×Hs−13(R+)×Hs−33(R+).

Multiple solutions of a Sturm-Liouville boundary value problem of nonlinear differential inclusion with nonlocal integral conditions

&lt;abstract&gt;&lt;p&gt;The existence of solutions for a Sturm-Liouville boundary value problem of a nonlinear differential inclusion with nonlocal integral condition is studied. The maximal and minimal solutions will be studied. The existence of multiple solutions of the nonhomogeneous Sturm-Liouville boundary value problem of differential equation with nonlocal integral condition is considered. The eigenvalues and eigenfunctions are investigated.&lt;/p&gt;&lt;/abstract&gt;

Analytical solution of a non-homogeneous boundary-value problem for the transport equation in an earth to air energy exchanger by initial base analysis method

In this paper, a one-dimensional time-dependent transport equation with inhomogeneous Dirichlet boundary conditions, often encountered in the study of an Earth to Air Energy Exchanger, is solved by a practical analytical method. The initial condition function is not randomly set to verify the transport equation at the initial time, but determined as a first approximation to the solution of the transport equation (at t=0). As in the homotopy analysis method, the solution of the transport equation is expressed by the set of a basis function constructed with the obtained initial condition function. The components of the solution in this base, are time’s functions deduced from the boundary conditions of the problem. Suitable endings are noticeable between the proposed model and experimental measurements.

ON A TOPOLOGICAL METHOD FOR CALCULATING THE INDEX OF QUASI-SINGULAR INTEGRAL EQUATION

A reduced spectral problem is considered for an elliptic type operator, in particular, for the Cauchy - Riemann operator with regular boundary value conditions to a quasi-singular integral equation with continuous kernel. Structure of the kernel of the quasi-singular integral equation is studied in explicit form. Index is calculated and condition for Noetherity of the investigated quasi-singular integral equation is established by the topological method on the complex plane. In this case, the spectral parameters are described under which the nonhomogeneous boundary value problem with shift for the Cauchy-Riemann equations is solvable everywhere in the class of continuous functions on the unit circle. The given problem is a nonlocal nature, and similar problems for the Cauchy-Riemann operation are described by M. Otelbaev in 1982. The solution of some singular integral equations with kernels depending on the difference and the sum of the arguments were studied in the works of I.I. Kalmushevsky. This interest is explained both by the theoretical significance of the results obtained and by the possibilities of important applications. Similar boundary value problems arise in the mathematical modeling of gas dynamics problems, soil moisture prediction, radiation transfer, population genetics, etc. The simplest examples of these boundary conditions were formulated by V.A. Steklov 1922. In this paper, the fundamental difference from the above work is the topological approach to calculating the index and establishing the Noether condition on the complexs plane.