Boundary Value System Research Articles

Multiple solutions for systems of multi-point boundary value problems

In this paper, we establish the existence of at least three solutions of the multi-point boundary value system

Exponential spline approach for the solution of fourth order obstacle boundary value problems

An exponential quintic spline technique at mid knots is developed for approximating the solution of system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. The present technique gives a class of methods of order two, four and six. Two numerical examples are considered for the numerical illustration of the proposed method. It is shown that the method developed in this paper is more efficient than the other finite difference, collocation and spline methods.

Infinitely Many Weak Solutions for a Class of Two-point Boundary Value Systems

In this paper, we prove the existence of an open interval $]\lambda^{'},\lambda^{''}[$ for each $\lambda$ of which a class of two-point boundary value equations depending on $\lambda$ admits infinitely many weak solutions. The approach is based on variational methods.

Multiple Solutions for a Class of Multipoint Boundary Value Systems Driven by a One‐Dimensional (p1, …, pn)‐Laplacian Operator

Employing a recent three critical points theorem due to Bonanno and Marano (2010), the existence of at least three solutions for the following multipoint boundary value system in (0,1), , for 1 ≤ i ≤ n, is established.

Positive Solutions for p‐Laplacian Fourth‐Order Differential System with Integral Boundary Conditions

This paper investigates the existence of positive solutions for a class of singular p‐Laplacian fourth‐order differential equations with integral boundary conditions. By using the fixed point theory in cones, explicit range for λ and μ is derived such that for any λ and μ lie in their respective interval, the existence of at least one positive solution to the boundary value system is guaranteed.

A critical points approach to multiplicity results for multi-point boundary value problems

In this article, we prove the existence of at least three classical solutions to the multi-point boundary value system for i = 1, … , n. Our main tool is a recent three critical points theorem of B. Ricceri.

A symbolic computation system for the calculus of moving surfaces

The calculus of moving surfaces (CMS) provides analytic tools for finding solutions to a wide range of problems with moving surfaces including fluid film dynamics, boundary variation problems, and shape optimization problems. The CMS is an extension of tensor calculus on stationary surfaces to moving surfaces. As with any analytic framework, the complexity of calculations grows rapidly with the order of approximation. This quickly causes problems to become complex enough that hand calculations become error prone or intractable. A symbolic computation system will alleviate these problems, allowing researchers to examine problems that have not been previously solvable. No symbolic calculus system is currently available that supports the CMS. We have developed a prototype symbolic computation system that can solve boundary variation problems with the help of the CMS. Our system has been used to solve a series of model problems of interest to applied mathematicians. The motivation is a boundary problem proposed by Grinfeld and Strang [2] in 2004. What is the series in 1/N for the simple Laplace eigenvalues λN on a regular polygon with N sides? In [2], the idea of expressing λN,n as a series in 1/N was put forth and in [3] the first several terms were computed using the calculus of moving surfaces. The prototype implementation was successfully used to find an error in the fourth term in the series expansion in [3] which was previously computed by hand. The CMS approach to this problem is essentially the same as for Poisson’s equation on polygons, which is used as the model problem in [1]. This problem involves Poisson’s equation ∇i∇u = 1 and is therefore simpler than the eigenvalue equation ∇i∇u = −λu that is at the heart of the model problem. The fundamental simplification is unrelated to the CMS: it comes from the fact that all solution variations un (where n is the order of the variation) satisfy Laplace’s equation in the interior. This makes it easy to solve for un and use the result in the next order of variation. The expression for E1 and the equation for u1 are obtained analytically. Higher order variations follow by direct application of the rules of CMS. The second order variation u2 is governed by the boundary value system

Existence of a positive solution to systems of differential equations of fractional order

In this paper, we consider a system of (continuous) fractional boundary value problems given by { − D 0 + ν 1 y 1 ( t ) = λ 1 a 1 ( t ) f ( y 1 ( t ) , y 2 ( t ) ) , − D 0 + ν 2 y 2 ( t ) = λ 2 a 2 ( t ) g ( y 1 ( t ) , y 2 ( t ) ) , where ν 1 , ν 2 ∈ ( n − 1 , n ] for n > 3 and n ∈ N , subject either to the boundary conditions y 1 ( i ) ( 0 ) = 0 = y 2 ( i ) ( 0 ) , for 0 ≤ i ≤ n − 2 , and [ D 0 + α y 1 ( t ) ] t = 1 = 0 = [ D 0 + α y 2 ( t ) ] t = 1 , for 1 ≤ α ≤ n − 2 , or y 1 ( i ) ( 0 ) = 0 = y 2 ( i ) ( 0 ) , for 0 ≤ i ≤ n − 2 , and [ D 0 + α y 1 ( t ) ] t = 1 = ϕ 1 ( y ) , for 1 ≤ α ≤ n − 2 , and [ D 0 + α y 2 ( t ) ] t = 1 = ϕ 2 ( y ) , for 1 ≤ α ≤ n − 2 . In the latter case, the continuous functionals ϕ 1 , ϕ 2 : C ( [ 0 , 1 ] ) → R represent nonlocal boundary conditions. We provide conditions on the nonlinearities f and g , the nonlocal functionals ϕ 1 and ϕ 2 , and the eigenvalues λ 1 and λ 2 such that the system exhibits at least one positive solution. Our results here generalize some recent results on both scalar fractional boundary value problems and systems of fractional boundary value problems, and we provide two explicit numerical examples to illustrate the generalizations that our results afford.

Boundary shape control of the Navier-Stokes equations and applications

In this paper, the geometrical design for the blade’s surface % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Wqpe0xc9q8qqaqFn0dXdir-xcvk9pIe9q8qq % aq-dir-f0-yqaqVeLsFr0-vr0-vr0db8meaabaqaciGacaGaaeqaba % Waaqaafaaakeaarmqr1ngBPrgitLxBI9gBaGGbaiab-fribdaa!44D6! $$ \Im $$ in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade’s surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.

Multiplicity results for a class of gradient systems depending on two parameters

In this paper we prove the existence of at least three solutions for a class of two-point boundary value systems of the type { − u i ″ + u i = λ F u i ( x , u 1 , … , u n ) + μ G u i ( x , u 1 , … , u n ) , u i ′ ( a ) = u i ′ ( b ) = 0 for 1 ≤ i ≤ n . The approach is fully based on a recent three critical points theorem of Ricceri [B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (9) (2009) 3084–3089].

Double barrier option under regime-switching exponential mean-reverting process

In this paper, we study a double barrier option when the underlying asset price follows a regime-switching exponential mean-reverting process. Our method is a combination of analysis of a deterministic boundary value problem with a probabilistic approach. In this setting, the double barrier option prices satisfy a system of m linear second-order differential equations with variable coefficients and with Dirichlet boundary conditions, where m is the number of regimes considered for the economy. We prove the existence of a smooth solution of the boundary value system by the method of upper and lower solutions; we proceed to construct monotonic sequences of upper and lower solutions that converge to true solutions, respectively. The uniqueness of the solution is established by applying Dynkin's formula. This proof by construction also provides a numerical procedure to compute approximate option values. An important feature of the proposed numerical method is that the true option values are bracketed by the upper and the lower solutions. Examples are provided to illustrate the method.

Positive solutions for a nonlinear second-order semipositone boundary value system

In this paper, by using the fixed point theorem in cones, the existence of positive solutions for systems of nonlinear second-order semipositone boundary value system { x ″ ( t ) + λ f 1 ( t , x ( t ) , y ( t ) ) + μ g 1 ( t , x ( t ) , y ( t ) ) = 0 , y ″ ( t ) + λ f 2 ( t , x ( t ) , y ( t ) ) + μ g 2 ( t , x ( t ) , y ( t ) ) = 0 , x ( 0 ) = x ( 1 ) = y ( 0 ) = y ( 1 ) = 0 , 0 ≤ t ≤ 1 , which nonlinearity f i and g i ( i = 1 , 2 ) are both semipositone, is obtained under the condition that f i ( i = 1 , 2 ) and g i ( i = 1 , 2 ) are suplinear or sublinear. The results obtained in this paper essentially improve and generalize many well-known results.

Imposing FIR Structure on $H^2$ Preview Tracking and Smoothing Solutions

This paper studies continuous-time $H^2$ feedforward tracking and estimation problems with FIR (finite impulse response) solutions. Both causal and noncausal solutions are considered. It is shown that these problems can be treated as special cases of static problems in the lifted domain, which, in turn, can be solved in terms of a two-point boundary value (Hamilton-Jacobi) system. This results in fixed-dimension closed-form solutions. A numerically stable form of these solutions, which uses matrix exponentials of Hurwitz matrices only, is also derived.

Positive solutions of n-order and m-order multi-point singular boundary value system

In this paper, we study the existence of positive solutions for the nonlinear n-order and m-order multi-point singular boundary value system. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear singular boundary value system are obtained.

The existence of positive solutions for nonlinear singular boundary value system with p-Laplacian

In this paper, we study the existence of positive solutions for the following nonlinear singular boundary value problem with p-Laplacian: ( ϕ p ( u ′ ) ) ′ + a ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , α ϕ p ( u ( 0 ) ) - β ϕ p ( u ′ ( 0 ) ) = 0 , γ ϕ p ( u ( 1 ) ) + δ ϕ p ( u ′ ( 1 ) ) = 0 , where ϕ p ( s) = ∣ s∣ p−2 s, p > 1, f is lower semi-continuous functions. By using the fixed-point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for nonlinear singular boundary value problem p-Laplacian are obtained.

The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator

In this paper, we study the existence of countable many positive solutions for a class of nonlinear singular boundary value systems with p-Laplacian operator: { ( ϕ p 1 ( u ′ ) ) ′ + a 1 ( t ) f ( u , v ) = 0 , 0 < t < 1 , ( ϕ p 2 ( v ′ ) ) ′ + a 2 ( t ) g ( u , v ) = 0 , 0 < t < 1 , { α 1 ϕ p 1 ( u ( 0 ) ) − β 1 ϕ p 1 ( u ′ ( 0 ) ) = 0 , γ 1 ϕ p 1 ( u ( 1 ) ) + δ 1 ϕ p 1 ( u ′ ( 1 ) ) = 0 , α 2 ϕ p 2 ( v ( 0 ) ) − β 2 ϕ p 2 ( v ′ ( 0 ) ) = 0 , γ 2 ϕ p 2 ( v ( 1 ) ) + δ 2 ϕ p 2 ( v ′ ( 1 ) ) = 0 , where ϕ p i ( s ) = | s | p i − 2 s , p i > 1 , f, g are lower semi-continuous functions, a i ( t ) has countable many singularities on ( 0 , 1 2 ) , i = 1 , 2 . By using the fixed-point theorem of cone expansion and compression of norm type, the existence of countable many positive solutions for nonlinear singular boundary value system with p-Laplacian operator are obtained.

Fast analytical short‐circuit current calculation of rectifier‐fed auxiliary subsystems

AbstractThe course of time of a three‐phase rectifier system which its alternating valve participation can be interpreted as a continuous sequence of alternating switching states. To allow a more convenient calculation, the substitutional circuit with the converter is transformed into state‐space coordinates. Hereby each operational mode of the rectifier can be represented by two linear independent space‐phasor component networks. In the state‐space, an analytical solution for this boundary value system can be carried out. After a retransformation back into the time domain, its time functions can be derived. In contrast to other calculation methods, no assumptions or simplifications have to be made like ideal smooth DC currents. Furthermore, all states of operation of the rectifier bridge can be easily calculated, which cover DC side idle‐running to DC short‐circuit.

Constructing eigenfunctions of strongly coupled parabolic boundary value systems

This paper deals with the study of existence conditions as well as the construction of eigenfunctions of strongly coupled parabolic boundary value partial differential systems.

Mesures de défaut de compacité, application au système de Lamé

We define the microlocal defect measures for boundary value systems satisfying the strong Lopatinski condition and we apply these notions to the study of the asymptotic propagation of the energy for the solutions of the Lamé system.

Existence theorems for a system of difference equations with ( n, p)-type conditions

We offer criteria for the existence of single and double solutions of fixed signs for the following system of boundary value problems Δ nu i(k)+q i(k,u 1(k),u 2(k),…,u m(k))=0, k∈{0,1,…,N}, Δ ju i(0)=0=Δ p i u i(N+n−p i), 0⩽j⩽n−2, i=1,2,…,m, where n⩾2, N⩾ n−1, 1⩽ p i ⩽ n−1, 1⩽ i⩽ m and the nonlinear terms q i 's can be `positive' or `nonpositive'.