Positive solutions for a system of higher-order multi-point boundary value problems

We present some results for positive solutions of a system of higher-order nonlinear ordinary differential equations, subject to multi-point boundary conditions.     

Linear and non-linear deflection analysis of thick rectangular plates—I. theoretical derivation

Non-linear analysis of thick rectangular plates has been conducted using general three-dimensional equations; simplifications are made only for small rotations. The governing dimensionless partial differential equations in terms of the displacement components U, V, and W were derived. Non-linear analysis using the von Kármán equations can be considered as a particular case when the plate thickness is very small. Finite difference techniques were used to transform the partial differential equations into an algebraic system of equations to yield the solutions. The successive approximation technique is used to pursue the convergence of the dimensionless non-linear deflection of thick plates, using an under-relaxation factor γ with the linear deflection results as the first guess for the analysis.     

The use of adaptive grid refinement for badly behaved elliptic partial differential equations

Adaptive grid refinement is potentially a very powerful means of dealing with singularities and other types of misbehavior in the solutions of elliptic partial differential equations. Combined with the multi-level iterative technique for solving the matrix equations, the method can be implemented in a reasonably efficient fashion.     

On the singular tracking problem

In this paper we continue the analysis of the problem of output tracking in the presence of singularities, whose study was begun by R. Hirschorn and J. Davis. We introduce further structure which is important in quantifying the multiplicity and smoothness of solutions to the problem. The paper is motivated by the analysis of those singular ordinary differential equations whose structure ultimately governs solutions to the singular tracking problem. In the particular case of time-varying linear systems, it is shown how the structure of their solutions in the case of regular and irregular singularities affects solutions to the tracking problem. Less specific results are also obtained in the full nonlinear case. P. E. Crouch and I. Ighneiwa were partially supported by N.S.F. Contract No. ECS 8703615.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions

In the this paper, we establish sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann-Liouville sense. Our analysis rely on Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results.     

A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations

This paper studies the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order q∈(1,2]. Our results are based on some standard fixed point theorems. Some illustrative examples are also discussed.     

Theorems on multidimensional laplace transform for solution of boundary value problems

Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of multidimensional Laplace transform. The general theory for obtaining solutions in this technique is developed in this paper by providing theorems on Laplace transform in n dimensions. Examples are presented for each theorem. Once the basic theorems are established it is possible to derive many useful transform pairs in n variables. Use of the above technique is illustrated by solution of an electrostatic potential problem.     

Shapes of liquid drops obtained using symbolic computation

The first aim of this paper is to show how two free boundary problems arising from fluid mechanics can be solved with a domain perturbation method. The second aim is to analyse the range of validity of the series solutions. The analysis will aim at identifying the location and the nature of the singularities characterizing these series. After expansion, the equations obtained are linear, but at each stage the length of the expressions grows exponentially. Herein we implement techniques for the automatic generation of hierarchical expression sequences and we present several tools for reducing the combinatorial blow-up of the expressions arising in these two problems.     

Numerical method of bicharacteristics for hyperbolic partial functional differential equations

Classical solutions of mixed problems for first-order partial functional differential equations in several independent variables are approximated in this paper by solutions of a difference problem of Euler type. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. Convergence of explicit difference schemes is proved by consistency and stability arguments. It is assumed that the given functions satisfy nonlinear estimates of Perron type. Differential equations with deviated variables and differential integral equations can be obtained from the general model by specifying the given operators.      

Legendre wavelets method for approximate solution of fractional-order differential equations under multi-point boundary conditions

In this paper, Legendre wavelet collocation method is applied for numerical solutions of the fractional-order differential equations subject to multi-point boundary conditions. The explicit formula of fractional integral of a single Legendre wavelet is derived from the definition by means of the shifted Legendre polynomial. The proposed method is very convenient for solving fractional-order multi-point boundary conditions, since the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. Several numerical examples are solved to demonstrate the validity and applicability of the presented method.     

A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations

We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second-order coercive linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic ‘energy’ reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine ‘truth-mesh’ discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid-flux candidate Lagrange multipliers generated by a K-element coarser ‘working-mesh’ approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomain-local, symmetric Neumann problems. The technique is illustrated for the convection-diffusion and linear elasticity equations.     

Hermite–Sobolev orthogonal functions and spectral methods for second- and fourth-order problems on unbounded domains

Hermite spectral methods using Sobolev orthogonal/biorthogonal basis functions for solving second and fourth-order differential equations on unbounded domains are proposed. Some Hermite–Sobolev orthogonal/biorthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. The convergence is analyzed and some numerical results are presented to illustrate the effectiveness and the spectral accuracy of this approach.     

Boundary stabilisation of the wave equation in the presence of singularities

We study the boundary stabilisation of the wave equation by a nonlinear feedback active on a part of the boundary in geometric situations for which the solutions have singularities. These singularities appear at the interfaces at which the mixed Neumann–Dirichlet boundary conditions meet. Under a simple geometrical condition concerning the orientation of the boundary, we obtain sharp energy decay rates under a general growth assumption on the feedback. We show that the singularities do not affect the energy decay rates and give examples.     

Numerical study for two-point boundary value problems using Green''s functions

In this paper, we study the Green's function to find numerical solutions of second-order ordinary differential equations for two-point boundary value problems. We derive some properties of Green's function which can be applied to a Green's function integral formula. And we discuss and analyze numerical solutions which are obtained by the Green's function method and a shooting method.     

Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions

This paper studies the existence of solutions for nonlinear fractional differential equations and inclusions of order q∈(3,4] with anti-periodic boundary conditions. In the case of inclusion problem, the existence results are established for convex as well as nonconvex multivalued maps. Our results are based on some fixed point theorems, Leray-Schauder degree theory, and nonlinear alternative of Leray-Schauder type. Some illustrative examples are discussed.     

Adaptive r and h — r algorithms for boundary elements

The techniques of mesh redistribution (r) and mesh refinement-redistribution (h – r) have been proven to be of considerable practical importance in the numerical solution of boundary value problems. In this paper, the implementation of these techniques in the boundary element formulation is considered. To assess the validity of the proposed algorithms, two model problems are employed. These problems are the potential equation and the equations of linear elasticity in two dimensions, both containing boundary singularities (cracks).     

Finite element computation of absorbing boundary conditions for time-harmonic wave problems

This paper proposes a new method, in the frequency domain, to define absorbing boundary conditions for general two-dimensional problems. The main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general. It is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software. Boundary conditions at various orders of accuracy can be obtained in a simple way. This is then applied to study some examples for which analytical or numerical results are available. Good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method.     

一类内部点级联的PDE-ODE系统的边界控制

本文运用backstepping方法研究了一类偏微分方程与常微分方程(PDE-ODE)级联系统的能稳性.常见的级联系统在边界点x=0处级联,而本文所讨论的级联系统在内部点x0∈(0,1)处级联,级联点的改变使得新系统的控制问题更加复杂.针对新系统,首先,我们改进了backstepping方法中的常见变换,改进后的变换与常见变换相比,增加了变换中的核函数,且得到的是带有多个相容性条件的核方程组,给求解带来了困难.文中运用了一系列的技巧解出核函数,从而得到反馈控制器;其次,运用同样的方法找到改进变换的逆变换;最后,选择合适范数,利用变换的有界性证明得到闭环系统的稳定性.     

Boundary value problems for a class of impulsive functional equations

This paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions. We study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times, by the use of the monotone method. Some of the options included in this formulation are differential equations with maximum and integro-differential equations. In this paper, we also prove that the Lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval. The corresponding results are established for the impulsive case. The general formulation includes several types of functional dependence (delay equations, equations with maxima, integro-differential equations). Finally, we consider the case of functional dependence which is given by nonincreasing and bounded functions.