Two-point Boundary Conditions Research Articles

Existence and uniqueness of solutions for nonlinear impulsive differential equations with two-point and integral boundary conditions

In the present paper, a system of nonlinear impulsive differential equations with two-point and integral boundary conditions is investigated. Theorems on the existence and uniqueness of a solution are established under some sufficient conditions on nonlinear terms. A simple example of application of the main result of this paper is presented.

О двукратной полноте собственных функций сильно нерегулярного квадратичного пучка дифференциальных операторов второго порядка

Saratov State Law Academy, Russia, 410056, Saratov, Volskaya st., 1, Oksana_Parfilova@mail.ruA class of strongly irregular pencils of ordinary differential operators of second order with constant coefficients is considered. Theroots of the characteristic equation of the pencils from this class are supposed to lie on a straight line coming through the originand on the both side of the origin. Exact interval on which the system of eigenfunctions is 2-fold complete in the space of squaresummable functions is finded.Key words: quadratic pencil, second order pencil, pencil of ordinary differential operators, two-point boundary conditions,homogeneous differential expression with constant coefficients, completeness of the system of eigenfunctions, non-completenessof the system of eigenfunctions.

Application of nonclassical separation of variables to a nonlocal heat problem

We consider an initial-boundary value problem for the heat equation with a nonlocal two-point boundary condition containing a parameter. By separating the variables in an auxiliary function system, we construct a regular solution. We obtain sufficient conditions for the absolute and uniform convergence of the series in the auxiliary system. We prove conditions close to necessary ones for the existence of a regular solution of the initial-boundary value problem.

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply <svg style="vertical-align:-2.3205pt;width:52.4375px;" id="M1" height="15.0875" version="1.1" viewBox="0 0 52.4375 15.0875" width="52.4375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153&#xA;l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g><g transform="matrix(.017,-0,0,-.017,24.575,12.138)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,38.327,12.138)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,46.486,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters <svg style="vertical-align:-2.3205pt;width:73.425003px;" id="M2" height="15.1125" version="1.1" viewBox="0 0 73.425003 15.1125" width="73.425003" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.162)"><path id="x1D703" d="M475 507q0 -83 -20 -172t-56 -167.5t-93.5 -129t-125.5 -50.5q-157 0 -157 227q0 78 21.5 164t59 161t96.5 123.5t126 48.5q79 0 114 -58t35 -147zM391 522q0 155 -81 155q-62 0 -111 -82.5t-73 -200.5h253q12 81 12 128zM373 346h-255q-12 -91 -12 -150q0 -72 20 -123&#xA;t63 -51q34 0 64 28.5t52.5 77t39 103t28.5 115.5z" /></g><g transform="matrix(.017,-0,0,-.017,14.409,12.162)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,21.107,12.162)"><path id="x1D717" d="M501 444q0 -256 -148 -397q-62 -59 -134 -59q-54 0 -88 36.5t-29 124.5q3 51 -6.5 70.5t-29.5 19.5q-14 0 -37 -7l-6 25q42 42 91 42q27 0 42 -21.5t18.5 -48t3.5 -68.5q0 -120 80 -120q127 0 162 337q-23 -17 -60 -32t-67 -15q-87 0 -136 49.5t-49 121.5q0 86 63.5 148&#xA;t149.5 62q75 0 127.5 -65t52.5 -203zM423 443q0 106 -42 167t-95 61t-79.5 -38t-26.5 -96q0 -69 43.5 -110.5t111.5 -41.5q52 0 86 27q2 20 2 31z" /></g><g transform="matrix(.017,-0,0,-.017,34.638,12.162)"><path id="x3E" d="M512 230l-437 -233v58l378 199v2l-378 200v58l437 -233v-51z" /></g><g transform="matrix(.017,-0,0,-.017,49.342,12.162)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,59.32,12.162)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,67.48,12.162)"><use xlink:href="#x29"/></g> </svg>, and the resulting equations together with the two-point boundary conditions constitute a system of <svg style="vertical-align:-2.3205pt;width:52.4375px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 52.4375 15.0875" width="52.4375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x1D441"/></g><g transform="matrix(.017,-0,0,-.017,24.575,12.138)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,38.327,12.138)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,46.486,12.138)"><use xlink:href="#x29"/></g> </svg> ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of <svg style="vertical-align:-2.3205pt;width:52.4375px;" id="M4" height="15.0875" version="1.1" viewBox="0 0 52.4375 15.0875" width="52.4375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x1D441"/></g><g transform="matrix(.017,-0,0,-.017,24.575,12.138)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,38.327,12.138)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,46.486,12.138)"><use xlink:href="#x29"/></g> </svg> ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms.

Computation of Green’s functions for boundary value problems with Mathematica

This paper is devoted to construct an algorithm that allows us to calculate the explicit expression of the Green’s function related to a nth-order linear ordinary differential equation, with constant coefficients, coupled with two-point linear boundary conditions. We develop this algorithm by making a Mathematica package.

A finite difference scheme for a fluid dynamic traffic flow model appended with two-point boundary condition

A fluid dynamic traffic flow model with a linear velocity-density closure relation is considered. The model reads as a quasi-linear first order hyperbolic partial differential equation (PDE) and in order to incorporate initial and boundary data the PDE is treated as an initial boundary value problem (IBVP). The derivation of a first order explicit finite difference scheme of the IBVP for two-point boundary condition is presented which is analogous to the well known Lax-Friedrichs scheme. The Lax-Friedrichs scheme for our model is not straight-forward to implement and one needs to employ a simultaneous physical constraint and stability condition. Therefore, a mathematical analysis is presented in order to establish the physical constraint and stability condition of the scheme. The finite difference scheme is implemented and the graphical presentation of numerical features of error estimation and rate of convergence is produced. Numerical simulation results verify some well understood qualitative behavior of traffic flow.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10307GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 43-52

ON SOLUTION OF A CLASS OF FUZZY BVPS

This paper investigates the existence and uniqueness of solutions to rst-order nonlinear boundary value problems (BVPs) involving fuzzy dif- ferential equations and two-point boundary conditions. Some sucient condi- tions are presented that guarantee the existence and uniqueness of solutions under the approach of Hukuhara dierentiability.

On the parametrization of boundary-value problems with two-point nonlinear boundary conditions

We obtain some results for the solutions of nonlinear boundary-value problems of a certain type with two-point nonlinear boundary conditions and reveal the efficiency of the procedure of reduction of the analyzed problem to a parametrized boundary-value problem with linear boundary conditions containing certain artificially introduced parameters. To study the transformed two-point problem, we propose a method based on the approximations of a special type constructed in the analytic form. It is shown that these approximations uniformly converge to a parametrized limit function and establish the relationship between this function and the exact solution. The proposed procedure leads to a certain system of algebraic equations whose solutions give numerical values of the parameters corresponding to the solution of the given two-point nonlinear boundary-value problem.

Two-Point Boundary Value Problems for a Class of Second-Order Ordinary Differential Equations

We study the general semilinear second-order ODE 𝑢   + 𝑔 ( 𝑡 , 𝑢 , 𝑢  ) = 0 under different two-point boundary conditions. Using the method of upper and lower solutions, we obtain an existence result. Moreover, under a growth condition on 𝑔 , we prove that the set of solutions of 𝑢   + 𝑔 ( 𝑡 , 𝑢 , 𝑢  ) = 0 is homeomorphic to the two-dimensional real space.

On Summability of Spectral Expansions Corresponding to the Sturm-Liouville Operator

We study the completeness property and the basis property of the root function system of the Sturm-Liouville operator defined on the segment [0, 1]. All possible types of two-point boundary conditions are considered.

Investigation of complex eigenvalues for a stationary problem with two-point nonlocal boundary condition

The Sturm–Liouville problem with one classical and another two-point nonlocal boundary condition is considered in this paper. These problems with nonlocal boundary condition are not self-adjoint, so the spectrum has complex points. We investigate how the spectrum in the complex plane of these problems (and for the Finite-Difference Schemes) depends on parameters γ and ξ of the nonlocal boundary conditions.

Exact Mathematical Model and its Numerical Solution of a Two-Layer Beam Subjected to Non-Uniform Temperature Rise

Based on accurately considering the axial extension, geometrically nonlinear governing equations for a two-layer beam subjected to thermal load were formulated. By using a shooting method, the strongly nonlinear ordinary differential equations with two-point boundary conditions were solved and numerical solution for thermal post-buckling and bending deformation of a two-layer beam with pinned-pinned ends and subjected to transversely non-uniform temperature rising were obtained. As an example, equilibrium paths and configuration for a beam laminated by brass and steel are presented and characteristic curves of the nonlinear deformation changing with the thermal load were plotted. Effects of the geometric and material parameters on the deformation of the beam were discussed and analyzed in detail. The theoretical analysis and numerical results show that the bending deformation and the stretching-bending coupling terms of beam subjected to uniform or non-uniform temperature rising can be produced because of the non-homogenous distribution of the material properties. The bending deformation resulted from transversely temperature rise is primary deformation when values of average temperature rise parameter is under critical temperature, however, the curves become the thermal post-buckling equilibrium paths with the increment of average temperature rise when values of average temperature rise parameter exceed critical temperature.

Synthesizing optimal trajectories subject to boundary conditions for linear controllable systems

We obtain analytic expressions for control functions that synthesize various families of optimal trajectories in a linear-quadratic problem subject to two-point boundary conditions. We give illustrative examples.

Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations

Positive solutions of systems of Hammerstein integral equations are studied by using the theory of the fixed-point index for compact maps defined on cones in Banach spaces. Criteria for the fixed-point index of the Hammerstein integral operators being 1 or 0 are given. These criteria are generalizations of previous results on a single Hammerstein integral operator. Some of criteria are new and involve the first eigenvalues of the corresponding systems of linear Hammerstein operators. The existence and estimates of the first eigenvalues are given. Applications are given to systems of fractional differential equations with two-point boundary conditions. The Green’s functions of the boundary value problems are derived and their useful properties are provided. As illustrations, the existence of nonzero positive solutions of two specific such boundary value problems is studied.

Computational methods for Generalized Sturmians basis

The computational techniques needed to generate a two-body Generalized Sturmian basis are described. These basis are obtained as a solution of the Schrödinger equation, with two-point boundary conditions. This equation includes two central potentials: A general auxiliary potential and a short-range generating potential. The auxiliary potential is, in general, long-range and it determines the asymptotic behavior of all the basis elements. The short-range generating potential rules the dynamics of the inner region. The energy is considered a fixed parameter, while the eigenvalues are the generalized charges. Although the finite differences scheme leads to a generalized eigenvalue matrix system, it cannot be solved by standard computational linear algebra packages. Therefore, we developed computational routines to calculate the basis with high accuracy and low computational time. The precise charge eigenvalues with more than 12 significant figures along with the corresponding wave functions can be computed on a single processor within seconds.

Upper and Lower Solutions for Regime-Switching Diffusions with Applications in Financial Mathematics

This paper develops a method of upper and lower solutions for a general system of second-order ordinary differential equations with two-point boundary conditions. Our motivation of study stems from a class of financial mathematics problems under regime-switching diffusion models. Two examples are double barrier option valuation and optimal selling rules in asset trading. We establish the existence of a unique $C^2$ solution of the two-point boundary value problem. We construct monotone sequences of upper and lower solutions that are shown to converge to the unique solution of the boundary value problem. This construction provides a feasible numerical method to compute approximate solutions. An important feature of the proposed numerical method is that the unique solution is bracketed by the upper and lower approximate solutions, which provide an interval estimate of the unique solution function. We apply the general results to a regime-switching mean-reverting model and improve related results already rep...

Existence of solutions to boundary value problems at full resonance

The focus of this paper is the study of nonlinear differential equations of the form ˙ xi(t )= ai(t)xi(t)+fi(e,t,x1(t),···,xn(t)), i = 1,2,···,n, subject to two-point boundary conditions bixi(0)+dixi(1 )= 0, i = 1,2,···,n. We formulate sufficient conditions for the existence of solutions based on the dimension of the solution space of the corresponding linear, homogeneous equation and the properties of the non- linear term when e = 0. We focus on the case when the solution space of the corresponding linear, homogeneous equation is n-dimensional; that is, when the system is at full resonance. The argument we use relies on the Lyapunov-Schmidt procedure and the Schauder fixed point theorem.

Nonlinear two-point boundary value problems on time scales

This work gives a criterion for the existence of positive solutions for nonlinear second order ordinary differential equations with two-point boundary conditions on time scales. Moreover, for some source terms which are in the sense of sublinear or superlinear, we also formulate corollaries and examples for applications and we improve previous results related with the first eigenvalue discussed by Professor Li.

Nodal solutions of boundary value problems with boundary conditions involving Riemann–Stieltjes integrals

We study the nonlinear boundary value problem consisting of the equation − y ″ = ∑ i = 1 m w i ( t ) f i ( y ) and a boundary condition involving a Riemann–Stieltjes integral. By relating it to the eigenvalues of the corresponding linear Sturm–Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. The shooting method and an energy function are used to prove the main results.

Nonlocal two-point problem for partial differential equations with linearly dependent coefficients

In the Cartesian product of the time segment and a spatial multidimensional torus, we investigate a problem with nonlocal two-point boundary conditions for a typeless partial differential equation of the second order with constant linearly dependent coefficients. Conditions for the unique solvability of the problem in the scale of Sobolev spaces are established. Metric theorems on estimates from below for small denominators on linear manifolds are proved.