Initial-boundary Value Research Articles

A Simple Introduction to Integral Equations

Integral equations are equations in which the unknown function appears inside a definite integral. They are closely related to differential equations. Initial value problems and boundary value problems for ordinary and partial differential equations can often be written as integral equations (see [7] for an introduction to the technique), and some integral equations can be written as initial or boundary value problems for differential equations. Problems that can be cast in both forms are generally more familiar as differential equations, owing to the larger collection of analytical procedures for solving differential equations. Many applications are best modeled with integral equations, but most of these problems require a lengthy derivation. A relatively simple example is the model for population dynamics, with birth and death rates that depend on age. This model was first formulated in 1922 by Alfred J. Lotka [8], who is best known for the LotkaVolterra predator-prey population model. A similar model has been used to model the spread of the HIV virus among IV drug users [4]. A simple age-dependent population model will be developed at the end of this article. Integral equations are also important in the theory and numerical analysis of differential equations; this is where the mathematics student is most likely to encounter them. For example, Picard's existence and uniqueness theorem for first-order initial value problems is conveniently proved using integral equations [1]; the proof is constructive and can be used to formulate a method for numerical solution of initial value problems. Systematic study of integral equations is usually undertaken as part of a course in functional analysis (see [6]) or applied mathematics (see [9]). This advanced setting is required for a full appreciation of integral equation theory, but it makes the subject accessible only to the advanced student. By contrast, several of the important results in the theory of integral equations can be demonstrated using nothing more than elementary analysis, and the elaboration of this statement is the goal of the present discussion. In fact, all but one of the results presented here will be derived using nothing more than the material presented in a standard advanced calculus course. This gives less advanced students of mathematics a chance to encounter some integral equation concepts that play an important role in the theory and application of continuous mathematics.

The errors estimation of the Chebyshev spectral-difference method for two-dimensional vorticity equation

We construct a Chebyshev spectral-difference scheme for solving two-dimensional vorticity equation with semi-homogeneous boundary conditions. We vigorously analyse the errors produced by initial errors and boundary value errors. The energy method is the main method used for errors estimation in this paper.

Steady Motion of a Thread Over a Rotating Roller

Dynamic equations of steady motion governing the behavior of threads over rotating arbitrary-axisymmetric rollers are derived. Various types of boundary conditions resulting in initial value or boundary value problems are discussed. Analytical solutions for the case of a circular cylinder are found. Two of the integrals obtained are exact. The third one, being a perturbation result, is thus approximate. Comparisons of results for a circular cylinder with those for tapered and parabolic rollers are made.

Microwave Heating of Materials with Nonohmic Conductance

The microwave heating of a material with temperature-dependent, nonohmic con-ductance is considered both analytically and numerically. In the case when the microwave amplitude is small, it is shown using a multiple scales expansion that the heating is governed by a Ginzburg–Landau type equation. This equation does not possess the solitary wave solutions of the full Ginzburg–Landau equation. Approximate solutions in the form of a slowly varying soliton and a front are found in certain parameter limits; these solutions compare very well with numerical solutions of the full governing equations. Initial-boundary value and initial value problems are considered numerically with particular emphasis on the structure of fronts.

Nilpotent and recursive flows

In this paper we introduce a class of nonlinear vector fields on infinite dimensional manifolds such that the corresponding evolution equations can be solved with the same method one uses to solve ordinary differential equations with constant coeficients. Mostly, these equations are nonlinear partial differential equations. It is shown that these flows are characterized by a generalization of the ‘method of variation of constants’ which is widely used for second order problems to find general solutions out of particular ones. Invariant densities are constructed for these flows in a natural way. These invariant densities are providing an essential tool for solving initial value and boundary value problems for the equations under consideration. Many applications are presented

Continuous dependence and differentiation of solutions of finite difference equations

Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the nth order finite difference equation, u(m + n) = f(m, u(m), u(m + 1), …, u(m + n − 1)), m ∈ ℤ.

Initial Boundary-Value Problems of Nonlinear Wave Equations in an Exterior Domain

In this paper we investigate the existence and uniqueness of the global solutions to the initial boundaryvalue problems of nonlinear wave equations in an exterior domain. When the space dimension n is at least 3, the unique global solution of the above problem is obtained for small initial data, even if the nonlinear term is fully nonlinear and contains the unknown function itself.

Existence of solutions to some initial value, two-point and multi-point boundary value problems with discontinuous nonlinearities

A variety of initial and boundary value problems in applied mathematics have nonlinear, discontinuous forcing terms. Existence theorems for problems of this sort are established for initial value, two-point, and multi-point boundary value problems for ordinary differential equations. Applications to steady-state heat conduction problems are given

Solution of linear dynamic systems with initial or boundary value conditions by shifted Chebyshev approximations

The shifted Chebyshev polynomial approximation is employed to solve the linear, constant parameter, ordinary differential equations of initial or two-point boundary value problems. An effective recursive algorithm is developed to calculate the expansion coefficients of the shifted Chebyshev series. An effective transformation is proposed to transform the two-point boundary value problem into an initial value problem. An illustrative example is included to show that the computational results are accurate.

A series solution algorithm for initial boundary-value problems with variable properties

A multiple infinite trigonometric cum polynomial series method for solving initial-boundary value problems governed by hyperbolic differential equations with variable coefficients is developed. The method proposed herein can be easily applied to a broad class of engineering systems including those cases where boundary conditions may vary with time. In the proposed mathematical technique, the solution form is assumed as a combination of infinite Fourier series and polynomial series of nth order, where n is the order of the differential equation. The coefficients of the polynomial series are obtained as functions of undetermined Fourier series coefficients by satisfying the initial-boundary conditions. The variable coefficients are expanded in appropriate half-range sine or cosine series. Insertion of the above Fourier-polynomial series solutions into the differential equation and application of orthogonality conditions leads to a linear summation equation which can be solved in open form. However, the authors have developed a closed-form series solution consisting of a highly efficient algorithm. The major advantage of this technique is the development of a solution algorithm, coupled with the multiple infinite trigonometric cum polynomial series solutions, leading to fast converging series solutions. A representative initial and boundary value problem governed by hyperbolic partial differential equations of variable coefficients is presented herein to demonstrate the efficiency and accuracy of the method.

The Burgers hierarchy: on nonlinear initial and boundary value problems

Systems of nonlinear equations of the Burgers-hierarchy type are reduced to linear canonical form via Backlund transformations with a view to the treatment of initial and nonlinear boundary value problems.

A priori estimates for the displacement and velocity vectors in the dynamical theory of linear viscoelasticity

Estimates for the solution of the initial mixed boundary value problem in the theory of linear viscoelasticity are obtained in terms of given data. The estimates imply that the solution of the considered problem is unique.

From kinematical geodesy to inertial positioning

WhenH. Moritz (1967, 1971) studied “kinematical geodesy” for the purpose of separation of gravitation and inertia, especially within combined accelerometer-gradiometer systems, it was hard to believe that within five years time inertial survey systems would be available, exactly operating according to his theoretical design. Here, we attempt to give a geodetic introduction into the fundamental equation of inertial positioning materialized by inertial survey systems with emphasis on a careful error model, including 36 parameters of type time interval, initial positions, initial gravity, varying acceleration, varying gravity gradients, accelerometer bias, accelerometer random uncertainty, accelerometer non-orthogonality, initial misalignment angles, accelerometer scale factor uncertainty. The notion of “multipoint” boundary value problem and initial value problem of inertial positioning is reviwed. So-called “post-mission” adjustment techniques for inertial surveys are discussed.

A Test of Moving Mesh Refinement for 2-D Scalar Hyperbolic Problems

We present an algorithm for numerically solving hyperbolic equations with moving shock fronts. The central idea is a moving, finer mesh which follows the shock. Computational results are presented.

Limit theorems for solutions of stochastic differential equation problems

In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.Limit theorems are proved for the eigenvalues and the eigenfunctions of eigenvalue problems and for the solutions of boundary value problems and initial value problems.

An iteration method for a nonlinear initial value problem arising from the kinetic theory of vehicular traffic

Abstract In the framework of the Paveri-Fontana model for dilute vehicular traffic in highways, an iteration method is provided which gives the car distribution function. This function, which satisfies a nonlinear initial-value boundary problem, is given as the limit of two sequences of solutions of linear equations with the same initial and boundary conditions.

Weak Solutions of $(p(x)u'(x))' + g(x)u'(x) + qu(x) = f$ with $q,f \in H_{ - 1} [a,b]$, $0 < p(x) \in L_\infty [a,b]$, $g(x) \in L_\infty [a,b]$ and $u \in H_1 [a,b

In this paper we consider weak solutions $u(x) \in H_1 [a,b]$ of the differential equation $(p(x)u'(x))' + g(x)u'(x) + qu(x) = f$ where q and f may be distributional derivatives of elements in $L_2 [a,b]$. The linear initial value and linear boundary value $B_1 u \equiv \alpha _1 u(a) + \beta _1 u'(a) = 0$, $B_2 u \equiv \alpha _2 u(b) + \beta _2 u'(b) = 0$, $\alpha _1 $, $\alpha _2 > 0$, $\beta _1 \leqq 0 \leqq \beta _2 $ problems are studied. Also the semilinear boundary value problem where $f = f(x,u)$ and $B_1 u = b_1 (u)$ and $B_2 u = b_2 (u)$ is studied.

A hereditary partial differential equation with applications in the theory of simple fluids

The linearized boundary-initial history value problem for simple fluids obeying the Coleman-Noll constitutive equation $$S + p\delta = 2\int\limits_0^\infty {m(s)(E(t - s} ) - E(t))ds$$ is considered. Here S is the stress tensor, δ the Kronecker delta, p the constitutively indeterminate mean normal stress, E the infinitesimal strain tensor, and m(s) a material function. The shear relaxation modulus G is defined as $$G(s) = \int\limits_\infty ^s {m(\xi )d\xi .}$$ (i) In this paper it is shown that if G satisfies the assumptions $$G \in C^2 [0,\infty ),{\text{ }}G(s) \to 0{\text{ as }}s \to \infty,$$ (i) $$( - 1)^k \frac{{d^k G(s)}}{{ds^k }} > 0,{\text{ }}k = 0,1,$$ (ii) $$G''(s) \geqq 0,$$ (iii) then the rest state of the fluid is stable in an appropriate “fading memory” norm. The additional assumption $$ - \int\limits_0^\infty {G'} (s)s^2 ds < \infty$$ (iv) yields asymptotic stability.

Newton’s Method Techniques for Singular Perturbations

In this paper, existence and uniqueness of solutions to several classes of singularly perturbed initial value and boundary value problems are proven by use of Newton’s method. Asymptotic expansions of solutions to some of these problems are developed and analyzed, and proof of their asymptotic correctness is shown based on Newton’s method. As a by-product some numerical algorithms for these solutions are obtained.

A unified approach to certain nonlinear initial value and boundary value problems

is an mth-order linear differential operator defined on some subset of LP(a, b) and Y is a lower-order nonlinear operator. This general theorem is then applied to three types of problems. In the first application we establish the existence of an absolutely continuous solution of the initial value problem,