Results For Differential Equations Research Articles

Effect of Mass per Unit Length on freely vibrating Simply Supported Rayleigh Beam

In this paper, free vibration characteristics of a uniform Rayleigh beam are studied using the differential transform method. The procedure entails transforming the partial differential equation governing the motion of the beam under consideration and the associated boundary conditions. The transformation yields a set of difference equations. Some simple algebraic operations are performed on the resulting difference equations to determine any ith natural frequency and the closed-form series function for any ith mode shape. Finally, one problem is presented to illustrate the implementation of the present method and analyse the effect of mass per length on the natural frequencies of the beam.

A new three-point sixth-order THAGE iteration method for mildly nonlinear two-point boundary value problems with engineering applications

We study a new sixth-order compact discretization using uniform three-grid point for the mildly nonlinear differential equation $$\phi ''=g(t,\phi )$$ , subject to the values of ϕ given at two end points of the regular solution interval. We also discuss three-step AGE (THAGE) iteration method as an application to the resulting difference equation as a powerful numerical computation device. In this algorithm, the common term is evaluated first to save the CPU time in comparison with the corresponding two-step algorithm. In addition, the error analysis is studied. Numerical performance is compared with the exact solution, and with the two-step AGE and SOR iteration methods.

Consensus and Information Cascades in Game-Theoretic Imitation Dynamics with Static and Dynamic Network Topologies

We construct a model of strategic imitation in an arbitrary network of players who interact through an additive game. Assuming a discrete time update, we show a condition under which the resulting difference equations converge to consensus. Two conjectures on general convergence are also discussed. We then consider the case where players not only may choose their strategies, but also affect their local topology. We show that for prisoner's dilemma, the graph structure converges to a set of disconnected cliques and strategic consensus occurs in each clique. Several examples from various matrix games are provided. A variation of the model is then used to create a simple model for the spreading of trends, or information cascades in (e.g., social) networks. We provide theoretical and empirical results on the trend-spreading model.

Malmquist-type Results for Difference Equations with Periodic Coefficients

An earlier result by N. Yanagihara leads us to consider the nature of a meromorphic solution $$f$$ of a difference equation $$\begin{aligned} f(z+c)=\sum _{j=0}^n p_j(z)(f(z))^j, \end{aligned}$$ where $$p_j$$ are periodic entire functions of the period $$c\in \mathbb {C}$$ , $$p_n \not \equiv 0$$ and $$n >1$$ . We shall show that if $$f$$ is non-periodic entire and of finite order of growth, it must be algebraic over any field that contains the coefficients of the difference equation. We also consider the special case $$n=2$$ more carefully and obtain specific information on the solution $$f$$ . Our methods are based on Nevanlinna theory and algebraic field theory.

Periodic Solutions of Certain Differential Equations with Piecewise Constant Argument

Existence criteria are derived for the eventually periodic solutions of a class of differential equations with piecewise constant argument whose solutions at consecutive integers satisfy nonlinear recurrence relations. The proof characterizes the initial values of periodic solutions in terms of the coefficients of the resulting difference equations. Sufficient conditions for the unboundedness, boundedness, and symmetry of general solutions also follow from the corresponding properties of the difference equations.

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

Oscillation Properties for Second-Order Half-Linear Dynamic Equations on Time Scales

Since Hilger (1990) introduced the theory of time scales, many authors have expounded on various aspects of this new theory; see the books (Bohner & Peterson, 2001, 2003) and the papers (Agarwal et al., 2007; Bohner & Saker, 2004; Chen, 2010; Chen & Liu, 2008; Doslý & Hilger 2002; Erbe et al., 2008; Hassan, 2008; Hassan, 2009; Karpuz, 2009; Medico & Kong, 2004; Saker, 2005; Zhang, 2011). A time scale T is an arbitrary nonempty closed subset of the reals R (see Hilger, 1990; Bohner & Peterson, 2001, 2003), and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential equations and of difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice–once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a time scale. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations unify the theories of differential equations and difference equations, but it is also able to extend these classical cases to cases “in between,” e.g., to the so-called q-difference equations. Dynamic equations on time scales have a lot of applications in population dynamics, quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. Bohner and Peterson (2001) summarizes and organizes much of time scale calculus. For advances of dynamic equations on time scales, we refer the reader to (Bohner & Peterson, 2003).

The discrete mean square of Dirichlet $L$-function at integral arguments

In this paper we shall make complete structural elucidation of the explicit formula for the (discrete) mean square of Dirichlet $L$-function at integral arguments, save for the case $s=1$, this being completely settled in [1] recently. We shall treat the cases of negative and positive integers arguments separately, the former case being a preliminary and inclusive in the second. It will turn out that in respective cases the characteristic difference properties of Bernoulli polynomials and of the Hurwitz zeta-function are essential and telescoping the resulting difference equations, we obtain the results, revealing the underlying simple structure (known before 1905 at least).

Linear Stability Analysis and Application of a New Solution Method of the Elastodynamic Equations Suitable for a Unified Fluid-Structure-Interaction Approach

In the conventional approach for fluid-structure-interaction problems, the fluid and solid components are treated separately and information is exchanged across their interface. According to the conventional terminology, the current numerical methods can be grouped in two major categories: partitioned methods and monolithic methods. Both methods use separate sets of equations for fluid and solid that have different unknown variables. A unified solution method has been presented in the previous work of Giannopapa and Papadakis (2004, “A New Formulation for Solids Suitable for a Unified Solution Method for Fluid-Structure Interaction Problems,” ASME PVP 2004, San Diego, CA, July, PVP Vol. 491–1, pp. 111–117), which is different from these methods. The new approach treats both fluid and solid as a single continuum; thus, the whole computational domain is treated as one entity discretized on a single grid. Its behavior is described by a single set of equations, which are solved fully implicitly. In this paper, the elastodynamic equations are reformulated so that they contain the same unknowns as the Navier–Stokes equations, namely, velocities and pressure. Two time marching and one spatial discretization scheme, widely used for fluid equations, are applied for the solution of the reformulated equations for solids. Using linear stability analysis, the accuracy and dissipation characteristics of the resulting difference equations are examined. The aforementioned schemes are applied to a transient structural problem (beam bending) and the results compare favorably with available analytic solutions and are consistent with the conclusions of the stability analysis. A parametric investigation using different meshes, time steps, and beam dimensions is also presented. For all cases examined, the numerical solution was stable and robust and therefore is suitable for the next stage of application to full fluid-structure-interaction problems.

Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme

We consider the two-dimensional time-dependent Schrödinger equation with the new compact nine-point scheme in space and the Crank-Nicolson difference scheme in time. For the resulting difference equation we derive discrete transparent boundary conditions in order to get highly accurate solutions for open boundary problems. Numerical experiments illustrate the perfect absorption of outgoing wave independently of their impact angle at the boundary. Finally, we apply inhomogeneous discrete transparent boundary conditions to the transient simulation of quantum waveguides.

An unsteady peristaltic transport phenomenon of non-Newtonian fluid – A generalised approach

Peristaltic transport of a non-Newtonian power law fluid characterised by the streaming blood through an axisymmetric tapered tubular vessel is studied under long wave length approximation. The flow is considered to be unsteady and two-dimensional so that both the axial and the radial velocity components of the fluid come into play. The generalised constitutive relations for the power law fluid are thoroughly encountered in the present analysis. The input pressure–radius relationship based on the assumption of the incompressibility and viscoelasticity of the vascular wall material in conformity to the experimental observations is given much attention in order to have resemblance to the in vivo situation. The nonlinear equations of motion governing the peristaltic transport phenomena are properly discretised with the use of suitable finite difference explicit scheme. The resulting difference equations are solved numerically by exploiting the appropriate initial and boundary conditions resulting from the physical and the biomechanical point of view. An extensive quantitative analysis is performed through numerical computations of the desired quantities of interest in order to estimate the effects of nonlinear inertial forces, the wall viscoelasticity, the tapering angle and the unsteadiness on the flow phenomena indicating their own role of importance in peristaltic transportation.

Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations

This paper is concerned with oscillation of second-order nonlinear neutral dynamic equations on time scales with a variable delay. By using the generalized Riccati technique and integral averaging technique, new oscillation criteria are obtained for all solutions of the equation. Some results extend known results for difference equations when the time scale is the set Z + of positive integers and for differential equations when the time scale is R . Several examples are given to illustrate the results of the paper.

OSCILLATION FOR NONAUTONOMOUS NEUTRAL DYNAMIC DELAY EQUATIONS ON TIME SCALES

The article is concerned with oscillation of nonautonomous neutral dynamic delay equations on time scales. Sufficient conditions are established for the existence of bounded positive solutions and for oscillation of all solutions of this equation. Some results extend known results for difference equations when the time scale is the set ℤ + of positive integers and for differential equations when the time scale is the set ℝ of real numbers.

On the road to chaos

Two simple mathematical models for how individual vehicles follow each other along a stretch of road are discussed. The resulting difference equations can be used as applications of techniques taught at A-level and first year undergraduate level, and as an introduction to the behaviour of the logistic map.

Local Stability of Euler's and Kahan's Methods

Two discretization methods, the forward Euler's method and the Kahan's reflexive method, are compared by looking at the local stabilities of fixed points of a system of differential equations. We explain why forward Euler's method is not as good from the viewpoint of complex analysis. Conformal mappings are used to relate the eigenvalues of the Jacobian matrices of the differential equations system and the resulting difference equations system. The Euler's method will not preserve Hopf bifurcation. The Kahan's method preserves the local stability of the fixed points of the differential equations.

Identifying a control function in two-dimensional parabolic inverse problems

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. This study presents numerical schemes for solving two two-dimensional parabolic inverse problems, i.e. for the determination of some unknown functions in the parabolic partial differential equations. These schemes are developed for determining the parameter p( t) which satisfy u t = u xx + u yy + p( t) u+ φ, in R×(0, T], u( x, y,0)= f( x, y),( x, y)∈ R=[0,1] 2. It is assumed that u is known on the boundary of R and subject to the integral overspecification over a portion of the spatial domain ∫ 0 1∫ 0 1u(x,y,t) dx dy=E(t) , 0⩽ t⩽ T, or to the overspecification at a point in the spatial domain u( x 0, y 0, t)= E( t), 0⩽ t⩽ T, where E( t) is known and ( x 0, y 0) is a given point of R. These schemes are considered for identifying the control parameter which produces, at any given time, a desired energy distribution in the spacial domain, or a desired temperature distribution at a given point in the spacial domain. The Saulyev’s finite difference techniques are used to compute the solution. Although the resulting difference equations do not appear explicit, a suitable use of the equations make them explicit. These explicit schemes are unconditionally stable. Numerical experiments are presented to demonstrate the accuracy of the finite difference schemes and the efficiency of the proposed computational procedures.

Positivity‐preserving, flux‐limited finite‐difference and finite‐element methods for reactive transport

AbstractA new class of positivity‐preserving, flux‐limited finite‐difference and Petrov–Galerkin (PG) finite‐element methods are devised for reactive transport problems.The methods are similar to classical TVD flux‐limited schemes with the main difference being that the flux‐limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite‐element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity‐preserving property. Analysis of the latter scheme shows that positivity‐preserving solutions of the resulting difference equations can only be guaranteed if the flux‐limited scheme is both implicit and satisfies an additional lower‐bound condition on time‐step size. We show that this condition also applies to standard Galerkin linear finite‐element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time‐step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd.

A new algorithm for the incorporation of arbitrary linear lumped networks into FDTD simulators

The inclusion of lumped elements, both linear and nonlinear, into the finite-difference time-domain (FDTD) algorithm has been recently made possible by the introduction of the lumped element FDTD method. Such a method, however, cannot efficiently and accurately account for two-terminal networks made of several lumped elements, arbitrarily connected together. This limitation can be removed as proposed in this paper by describing the network in terms of its impedance in the Laplace domain and by using appropriate digital signal-processing methodologies to fit the resulting description to Yee's algorithm. The resulting difference equations allow an arbitrary two-terminal network to be inserted into one FDTD cell, preserving the full explicit nature of the conventional FDTD scheme and requiring a minimum number of additional storage variables. The new approach has been validated by comparison with the exact solution of a parallel-plate waveguide loaded with lumped networks in the transverse plane.

The central two-point connection problem for the Heun class of ODEs

We present a numerical approach to the central two-point connection problem for the confluent cases of the Heun class of differential equations. The crucial step is an ansatz for the solutions of the equations in terms of a generalized power series (called Jaffe expansions). It is shown that the resulting difference equations for the coefficients of these series are of Poincare-Perron type. A (formal) asymptotic investigation of the solutions of these difference equations yields the exact eigenvalue condition.

A New Finite Variable Difference Method with Application to Locally Exact Numerical Scheme

A new finite variable difference method (FVDM) on finite differencing is presented. The essence of this method consists in determining the optimum spatial difference such that the total variance of the solution is a minimum under the condition that characteristic roots of the resulting difference equation are always nonnegative to ensure the numerical stability. The present FVDM is applied to the locally exact numerical scheme (LENS). The optimum spatial difference of the LENS is derived in terms of local mesh Reynolds numbers. By using this optimum spatial difference the numerical accuracy of the LENS for the linear convection–diffusion equations is increased without numerical oscillations for all mesh Reynolds numbers. The present study suggests that an optimum spatial difference from the viewpoint of numerical stability and accuracy exists according to the numerical schemes.