Dimensional Differential Equations Research Articles

Macroscopic energy diffusion for a chain of anharmonic oscillators

We study the energy diffusion in a chain of anharmonic oscillators where the Hamiltonian dynamics is perturbed by a local energy conserving noise. We prove that under diffusive rescaling of space-time, energy fluctuations diffuse and evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and some upper and lower bounds.

Formula omitted] solutions to backward stochastic differential equations with discontinuous generators

In this paper, we deal with the Lp(p>1) solutions to one dimensional backward stochastic differential equations (BSDEs) with discontinuous generators. We obtain an existence theorem of Lp solutions for BSDEs whose generators satisfy a kind of discontinuous condition in y and are uniformly continuous in z.

On the self-similarity of 1/fβ sequences synthesized by recursive filtering

This paper describes an approximate method for synthesizing sequences of statistically self-similar processes and analyses its performance to generate sample sequences with this statistical property. The method is based upon approximating the infinite dimensional difference equation which describes the FARIMA(0,α,0) model by a finite dimensional difference equation. The parameters estimation for parameterizing the binomial coefficients is performed by using deterministic signal modeling techniques. The three techniques considered are: Prony, Steiglitz MacBride, and Shaw methods. In addition to allow considerable savings in memory requirements and great reduction in computation time, the performance analysis results show that the generated sequences are statistically self-similar in the sense that the estimated Hurst parameter is very close to that imposed in the sequence generator.

Analysis of Direct Extrusion Operation Using The Bubnov-Galerkin Finite Element Model

This paper reports the analysis of direct extrusion operation using the Bubnov-Galerkin finite element model to get the pressure distributions along the cross-section of a blank. Four Lagrange quadratic elements were assembled to represent the blank. The governing equation is a one dimensional differential equation describing the pressure on the die-blank interface. The weighted residual form was obtained from the differential equation, the finite element model was obtained in a matrix form from the weighted residual, boundary condition were now applied to obtain the pressure distribution across the cross-section of the blank. Finite element results were obtained for a particular values of coefficient of friction and blank diameter and compared with the exact solution on a graph. Keywords: Direct extrusion operation, Bubnov-Galerkin residual scheme, finite element model, Lagrange quadratic element Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 435 – 442

Balanced POD for model reduction of linear PDE systems: convergence theory

We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3):997–1013, 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For exponentially stable systems with bounded finite rank input and output operators, we prove that the balanced realization can be expressed in terms of balancing modes, which are related to the Hankel operator. The balancing modes are required to be smooth, and this can cause computational difficulties for PDE systems. We show how this smoothness requirement can be lessened for parabolic systems, and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems. We prove entrywise convergence of the matrices in the approximate reduced order models in both cases, and present numerical results for two example PDE systems.

Multiplicity Results on Period Solutions to Higher Dimensional Differential Equations with Multiple Delays

This paper deals with the existence and multiplicity of periodic solutions to delay differential equations of the form $$\dot{z}(t)=-f(z(t-1))- f(z(t-2))-\cdots- f(z(t-2n+1)) $$ where \({z\in {\bf R}^N, f\in C({\bf R}^{N}, {\bf R}^N)}\). By using the S1 pseudo geometrical index theory in the critical point theory, some known results for Kaplan–Yorke type differential delay equations are generalized to higher dimensional case. As a result, the Kaplan–Yorke’s conjecture is proved to be true in the case of higher dimensional systems.

Higher Dimensional Differential Equations for Some Real World S imulation Processes and Solutions Thereof

Higher dimensional differential equations may express several real world simulation processes which depend upon their pre-history and subject to short-time disturbances. Such processes occur in the theory of optimal control, population dynamics, biotechnologies, economics, mathematical physics etc. So, the study of this class of dynamical systems is gradually gaining momentum. In the present work Avery-Peterson theorem has been envisaged for getting the positive periodic solutions of the corresponding differential equations for cones with impulses on time scales. By using the multiple fixed-point theorems we have shown through different lemmas and manipulation of several functions how the necessary criteria can be mathematically arrived at so that the results come to be feasible as well as effective.Keywords: Real world simulation processes; Theory of optimal control; Population dynamics; Biotechnologies; EconomicsThe Himalayan Physics Vol.2, No.2, May, 2011Page: 50-53Uploaded Date: 1 August, 2011

Asymptotic properties of fractional delay differential equations

In this paper we study the asymptotic properties of d -dimensional linear fractional differential equations with time delay. We present necessary and sufficient conditions for asymptotic stability of equations of this type using the inverse Laplace transform method and prove polynomial decay of stable solutions. Two examples illustrate the obtained analytical results.

A Numerical Study on the Pressure Variation in a Sudden Expansion and a Sudden Expansion with Central Restriction

In this paper, a numerical study on the performance of sudden expansion with central restriction, viewed as a diffuser, has been carried out. The two dimensional steady differential equations for conservation of mass and momentum are solved for Reynolds number(Re) from 10 to 100 and CR from 0% to 30% for an aspect ratio (AR) of 2 and fully developed velocity profile at inlet. From the study, it is revealed that for sudden expansion, maximum wall pressure decreases with the increase in Re. Sudden expansion with central restriction also behaves in same way. Location of the maximum wall pressure increases with increase in Re for both sudden expansion and sudden expansion with central restriction. The value of maximum average static pressure increases with increase in Re for both sudden expansion and sudden expansion with central restriction. Location of the maximum average static pressure from the throat, which may be considered as the effective length of the diffuser, increases with Re for both the cases. In case of sudden expansion with central restriction, maximum average static pressure increases with the increase in CR but its location from throat remains more or less same, at a particular value of Re. In comparison to a simple sudden expansion, the magnitude of maximum average static pressure is more and the location of maximum average static pressure from the throat is less for sudden expansion with central restriction. Therefore, choice of the configuration of sudden expansion with central restriction, as a diffuser, is more attractive in comparison to the configuration of a simple sudden expansion.

The Numerical Solutions of a Two-Dimensional Space-Time Riesz-Caputo Fractional Diffusion Equation

This paper is concerned with the numerical solutions of a two dimensional space-time fractional differential equation used to model the dynamic properties of complex systems governed by anomalous diffusion. The space-time fractional anomalous diffusion equation is defined by replacing second order space and first order time derivatives with Riesz and Caputo operators, respectively. By using Laplace and Fourier transforms, a general representation of analytical solution is obtained as Mittag-Leffler function. Gr\"{u}nwald-Letnikov (GL) approximation is also used to find numerical solution of the problem. Finally, simulation results for two examples illustrate the comparison of the analytical and numerical solutions and also validity of the GL approach to this problem.

The Numerical Solutions of a Two-Dimensional Space-Time Riesz-Caputo Fractional Diffusion Equation

This paper is concerned with the numerical solutions of a two dimensional space-time fractional differential equation used to model the dynamic properties of complex systems governed by anomalous diffusion. The space-time fractional anomalous diffusion equation is defined by replacing second order space and first order time derivatives with Riesz and Caputo operators, respectively. By using Laplace and Fourier transforms, a general representation of analytical solution is obtained as Mittag-Leffler function. Gr\"{u}nwald-Letnikov (GL) approximation is also used to find numerical solution of the problem. Finally, simulation results for two examples illustrate the comparison of the analytical and numerical solutions and also validity of the GL approach to this problem.

Skaičiuojamosios fluidų dinamikos modeliavimo taikymas kūginiam grįžtamojo srauto ciklonui tirti

Nagrinėjama ciklonų, skirtų kietosioms dalelėms atskirti iš oro srauto, efektyvumo problema. Apžvelgti eksperimentiniai ir teoriniai darbai. Pateiktos trimatės pernašos diferencialinės lygtys su Reinoldso įtempių turbulencijos ir kitais modeliais. Atliktas skaitinis oro srauto judėjimo modeliavimas kūginiame grįžtamojo srauto (KGS) ciklone, kurio aukštis – 0,50 m, cilindrinės dalies aukštis - 0,29 m, kūginės - 0,133 m, skersmuo - 0,20 m. Modeliuojant aprašytas oro srauto judėjimas ciklone taikant Navjė ir Stokso trimačių diferencialinių lygčių sistemą, diskretizuotą baigtiniais tūriais. Modeliavimo rezultatai, kai įtekėjimo greitis – 4,64 m/s ir debitas – 0,28 m3/s, neblogai sutapo su kitų autorių vykdytų eksperimentų rezultatais. Gauta vidutinė santykinė paklaida - ±5 proc.

Convergent snapshot algorithms for infinite-dimensional Lyapunov equations

We consider two algorithms to approximate the solution Z of a class of stable operator Lyapunov equations of the form AZ + ZA∗ + BB∗ = 0. The algorithms utilize time snapshots of solutions of certain linear infinite dimensional differential equations to construct the approximations. Matrix approximations of the operators A and B are not required and the algorithms are applicable as long as the rank of B is relatively small. The first algorithm produces an optimal low rank approximate solution using proper orthogonal decomposition. The second algorithm approximates the product of the solution with a few vectors and can be implemented with a minimal amount of storage. Both algorithms are known for the matrix case, however the extension of the algorithms to infinite dimensions appears to be new. We establish easily verifiable convergence theory and a priori error bounds for both algorithms and present numerical results for two model problems.

Domain of attraction analysis of a controlled hybrid reactor model

A domain of attraction (DOA) analysis has been performed for a simplified model of the primary circuit of a pressurized water nuclear power plant, the Paks Nuclear Power Plant in Hungary. A simple two dimensional autonomous hybrid differential equation model of the controlled primary circuit has been developed for the purpose of the stability analysis that contains both continuous and discrete variables. It has been shown that the DOA of the controlled system encapsulates the operating domain for any positive value of the considered feedback gain of the power controller and the reactor power. However, the gain has a great impact on the dynamics of the controlled system that is also investigated, and a practically advantageous domain of the gain has been determined.

Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations

We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some cellular neural networks, monotone systems, and semi-discretizations for hyperbolic conservation laws with a source term. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.

Transverse plane wave analysis of short elliptical chamber mufflers: An analytical approach

Short elliptical chamber mufflers are used often in the modern day automotive exhaust systems. The acoustic analysis of such short chamber mufflers is facilitated by considering a transverse plane wave propagation model along the major axis up to the low frequency limit. The one dimensional differential equation governing the transverse plane wave propagation in such short chambers is solved using the segmentation approaches which are inherently numerical schemes, wherein the transfer matrix relating the upstream state variables to the downstream variables is obtained. Analytical solution of the transverse plane wave model used to analyze such short chambers has not been reported in the literature so far. This present work is thus an attempt to fill up this lacuna, whereby Frobenius solution of the differential equation governing the transverse plane wave propagation is obtained. By taking a sufficient number of terms of the infinite series, an approximate analytical solution so obtained shows good convergence up to about 1300 Hz and also covers most of the range of muffler dimensions used in practice. The transmission loss (TL) performance of the muffler configurations computed by this analytical approach agrees excellently with that computed by the Matrizant approach used earlier by the authors, thereby offering a faster and more elegant alternate method to analyze short elliptical muffler configurations.

Stability of perturbed [formula omitted]-dimensional Volterra differential equations

In this work we deal with the problem of the stability and uniform stability of the perturbed n -dimensional Volterra integral and differential equation x ′ ( t ) = A ( t ) x ( t ) + ∫ − ∞ t C ( t , s ) x ( s ) d s + ∫ − ∞ t D ( t , s ) x ′ ( s ) d s + b ( t ) . Some sufficient conditions for the zero solution of this equation to be stable as well as uniformly stable have been obtained.

Bernstein Polynomials Solving One Dimensional Delay Volterra Integro Differential Equations

Bernstein Polynomials Solving One Dimensional Delay Volterra Integro Differential Equations

A maximum principle for relaxed stochastic control of linear SDEs with application to bond portfolio optimization

We study relaxed stochastic control problems where the state equation is a one dimensional linear stochastic differential equation with random and unbounded coefficients. The two main results are existence of an optimal relaxed control and necessary conditions for optimality in the form of a relaxed maximum principle. The main motivation is an optimal bond portfolio problem in a market where there exists a continuum of bonds and the portfolio weights are modeled as measure-valued processes on the set of times to maturity.

Numerical experiment of measurement-integrated simulation to reproduce turbulent flows with feedback loop to dynamically compensate the solution using real flow information

Reproduction of the exact structure of real turbulent flows is crucial in many applications. Four Dimensional variation (4D-VAR) is widely used in numerical weather forecasting, but it requires huge computational power to repeatedly solve flow dynamics and its adjoint, and, therefore, is not suitable to apply to problems of real-time flow reproduction such as feedback flow control. Kalman filter and observer, in which numerical solution converges to the real state asymptotically by means of the feedback signal proportional to the difference between the calculated state and the real state, requiring much less computational load than the variational method, are potential candidates to solve the problem. By comparing Kalman filter and observer, the latter has simpler structure retaining essential part of the state estimation. This study deals with a special type of observer, or measurement-integrated simulation (MI simulation), in which a SIMPLER-based flow solver is used as the mathematical model of the system in place of approximate small dimensional linear differential equations usually used in observers. Reproduction of the exact structure of a turbulent flow was investigated by a MI simulation. A numerical experiment was performed for a fully developed turbulent flow in a pipe with a square cross section. The MI simulation was performed with the feedback from the standard solution in the flow domain for the cases using: (1) all velocity components at all grid points, (2) partial velocity components at all grid points, or (3) all velocity components at partial grid points. Convergence of the MI simulation to the standard solution was investigated using the steady error norm for the convergent state and the time constant for the transient state. The result of the MI simulation using all the velocity information exponentially converges to the standard solution with a steady state error reduced from that of the ordinary simulation in a range of the feedback gain. Decreasing the feedback gain reduces the effect of feedback, and a feedback gain which is too large destabilizes the closed loop system, resulting in large error. The time constant decreases almost inversely proportional to the feedback gain as long as the feedback system is stable. For the MI simulation with the feedback using limited information, feedback using two velocity components by omitting one transverse velocity component showed a good result, although the other results were not satisfactory. For the MI simulation with the feedback using limited grid points, the result of the MI simulation applying the feedback at the grid points on every 20th plane in the x 1 direction was almost the same as that using all grid points at some feedback gain, while the result with the feedback on the planes skipped in the x 2 direction requires 10 times more planes to achieve the same reduction rate.