Analytic Solutions of a Second-Order Functional Differential Equation with a State Derivative Dependent Delay

We investigate an analytic solution of the second-order differential equation with a state derivative dependent delay of the form x″(z)=x(p(z)+bx′(z)). Considering a convergent power series g(z) of an auxiliary equation γ2g″(γz)g′(z)=[g(γ2z)-p(g(γz))]γg′(γz)(g′(z))2+p′′(g(z))(g′(z))3+γg′(γz)g″(z) with the relation p(z)+bx′(z)=g(γg-1(z)), we obtain an analytic solution x(z). Furthermore, we characterize a polynomial solution when p(z) is a polynomial

Closed-Form Solutions of the Thomas-Fermi in Heavy Atoms and the Langmuir-Blodgett in Current Flow ODEs in Mathematical Physics

Two kinds of second-order nonlinear, ordinary differential equations (ODEs) appearing in mathematical physics are analyzed in this paper. The first one concerns the Thomas-Fermi (TF) equation, while the second concerns the Langmuir-Blodgett (LB) equation in current flow. According to a mathematical methodology recently developed, the exact analytic solutions of both TF and LB ODEs are proposed. Both of these are nonlinear of the second order and by a series of admissible functional transformations are reduced to Abel’s equations of the second kind of the normal form. The closed form solutions of the TF and LB equations in the phase and physical plane are given. Finally a new interesting result has been obtained related to the derivative of the TF function at the limit

Predicting the approximate functional behaviour of physical systems

This dissertation addresses the problem of the computer prediction of the approximate behaviour of physical systems describable by ordinary differential equations.Previous approaches to behavioural prediction have either focused on an exact mathematical description or on a qualitative account. We advocate a middle ground: a representation more coarse than an exact mathematical solution yet more specific than a qualitative one. What is required is a mathematical expression, simpler than the exact solution, whose qualitative features mirror those of the actual solution and whose functional form captures the principal parameter relationships underlying the behaviour of the real system. We term such a representation an approximate functional solution.Approximate functional solutions are superior to qualitative descriptions because they reveal specific functional relationships, restore a quantitative time scale to a process and support more sophisticated comparative analysis queries. Moreover, they can be superior to exact mathematical solutions by emphasizing comprehensibility, adequacy and practical utility over precision.Two strategies for constructing approximate functional solutions are proposed. The first abstracts the original equation, predicts behaviour in the abstraction space and maps this back to the approximate functional level. Specifically, analytic abduction exploits qualitative simulation to predict the qualitative properties of the solution and uses this knowledge to guide the selection of a parameterized trial function which is then tuned with respect to the differential equation. In order to limit the complexity of a proposed approximate functional solution, and hence maintain its comprehensibility, back-of-the-envelope reasoning is used to simplify overly complex expressions in a magnitude extreme. If no function is recognised which matches the predicted behaviour, segment calculus is called upon to find a composite function built from known primitives and a set of operators. At the very least, segment calculus identifies a plausible structure for the form of the solution (e.g. that it is a composition of two unknown functions). Equation parsing capitalizes on this partial information to look for a set of termwise interactions which, when interpreted, expose a particular solution of the equation.The second, and more direct, strategy for constructing an approximate functional solution is embodied in the closed form approximation technique. This extends approximation methods to equations which lack a closed form solution. This involves solving the differential equation exactly, as an infinite series, and obtaining an approximate functional solution by constructing a closed form function whose Taylor series is close to that of the exact solutionThe above techniques dovetail together to achieve a style of reasoning closer to that of an engineer or physicist rather than a mathematician. The key difference being to sacrifice the goal of finding the correct solution of the differential equation in favour of finding an approximation which is adequate for the purpose to which the knowledge will be put. Applications to Intelligent Tutoring and Design Support Systems are suggested

Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties

[EN] In this paper we study random non-autonomous second order linear differential equations by taking advantage of the powerful theory of random difference equations. The coefficients are assumed to be stochastic processes, and the initial conditions are random variables both defined in a common underlying complete probability space. Under appropriate assumptions established on the data stochastic processes and on the random initial conditions, and using key results on difference equations, we prove the existence of an analytic stochastic process solution in the random mean square sense. Truncating the random series that defines the solution process, we are able to approximate the main statistical properties of the solution, such as the expectation and the variance. We also obtain error a priori bounds to construct reliable approximations of both statistical moments. We include a set of numerical examples to illustrate the main theoretical results established throughout the paper. 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Computational surface partial differential equations

Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties

Integral representations and Liouville theorems for solutions of periodic elliptic equations

The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arrives to Liouville type theorems, which describe the structure and dimension of the spaces of such solutions. The Liouville type theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M. Struwe for periodic second order elliptic equations in divergence form are significantly extended. Relations of these theorems with the analytic structure of the Fermi and Bloch surfaces are explained.Comment: 48 page