Remarks on a simple fractional-calculus approach to the solutions of the bessel differential equation of general order and some of its applications

In a remarkably large number of recent works, one can find the emphasis upon (and demonstrations of) the usefulness of fractional calculus operators in the derivation of (explicit) particular solutions of significantly general families of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to continue our investigation of this simple fractional-calculus approach to the solutions of the classical Bessel differential equation of general order and to show how it would lead naturally to several interesting consequences which include (for example) an alternative derivation of the complete power-series solutions obtainable usually by the Frobenius method. The underlying analysis presented here is based chiefly upon some of the general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations with polynomial coefficients.     

A generalized solution of an initial-boundary value problem arising in the mechanics of discrete-continuous systems

We define a generalized solution of an initial-boundary value problem for a linear system of differential equations with one ordinary differential equation and two partial differential equations (a hybrid system of differential equations). We prove that the problem is well-posed and has a unique generalized solution. An analytical formula for the solution is found. Such systems of differential equations arise in studying discrete-continuum mechanical systems.     

Feedback control of linear systems with distributed delay

The problem of optimal control of linear systems containing lumped delay, given by differential-difference equations, has been pursued by several authors. However, transportation-lags are better described by distributed delays giving systems that are described by a set of coupled partial and ordinary differential equations. The lumped part of the system is described by ordinary differential equations and the distributed part of the system is described by partial differential equations. The lumped as well as distributed parts are subject to control. The present paper discusses the control of such systems with quadratic performance measures. Riccati-like equations are derived and a technique for their numerical solution is presented.     

A new method for solving boundary value problems for partial differential equations

This paper proposes a symmetry–iteration hybrid algorithm for solving boundary value problems for partial differential equations. First, the multi-parameter symmetry is used to reduce the problem studied to a simpler initial value problem for ordinary differential equations. Then the variational iteration method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.     

Reduction of distributed system identification complexity using intelligent sensors

The identification problems of distributed-parameter systems (DPS) is dealt with, A special method of data collection by using moving sensors is proposed. The sensors are ‘intelligent’ in the sense that they are able to track the positions of the exiremum values of a system state at each time instant. It is shown how to reduce the identification problem for DPS to that of identifying a certain system described by ordinary differential equations (ODE). The proposed approach is applicable to the DPS described by quasi-linear partial differential equations of the hyperbolic type. Attention is focused on the problem of reduction, while the identification of the resulting ODE is not considered in detail, since it is an easier task.     

A difference scheme with high accuracy in time for fourth-order parabolic equations

A novel finite difference method is developed for the numerical solution of fourth-order parabolic partial differential equations in one and two space variables. The method is seen to evolve from a multiderivative method for second-order ordinary differential equations.The method is tested on three model problems, with constant coefficients and variable coefficients, which have appeared in the literature.     

Parameter identification in dynamic systems using the homotopy optimization approach

Identifying the parameters in a mathematical model governed by a system of ordinary differential equations is considered in this work. It is assumed that only partial state measurement is available from experiments, and that the parameters appear nonlinearly in the system equations. The problem of parameter identification is often posed as an optimization problem, and when deterministic methods are used for optimization, one often converges to a local minimum rather than the global minimum. To mitigate the problem of converging to local minima, a new approach is proposed for applying the homotopy technique to the problem of parameter identification. Several examples are used to demonstrate the effectiveness of the homotopy method for obtaining global minima, thereby successfully identifying the system parameters.     

A new energy harvester using a cross-ply cylindrical membrane shell integrated with PVDF layers

In this paper, we proposed a smart cylindrical membrane shell panel (SCMSP) model for vibration-based energy harvester. The SCMSP is made of an orthotropic elastic core covered by outer PVDF layers with transverse polarization vector. Electrodynamics governing equations of motion are derived by applying extended Hamilton’s principle. The governing equations are based on Donnell’s linear thin shell theory. The SCMSP displacement fields are expanded by means of double Fourier series satisfying immovable edges with free rotation boundary conditions and coupled system of linear partial differential equations are obtained. The discretized linear ordinary differential equations of motion are obtained using Galerkin method. The output power is taken as an indicating criterion for the generator. A parametric study for MEMS applications is conducted to predict the power generated due to radial harmonic ambient vibration. Optimal resistance value is also obtained for the particular electrode distribution that gives maximum output power. A low vibration amplitude (5?Pa), and a low-frequency (471.79?Hz) vibration source is targeted for the resonance operation, in which the output power of 0.4111?μW and peak-to-peak voltage of 0.2952?V are predicted.     

On testing the existence of universal denominators for partial differential and difference equations

We consider the problem of testing the existence of a universal denominator for partial differential or difference equations with polynomial coefficients and prove its algorithmic undecidability. This problem is closely related to finding rational function solutions in that the construction of a universal denominator is a part of the algorithms for finding solutions of such form for ordinary differential and difference equations.     

Solidification and control of a liquid metal column

In this paper, two different 1D mechanistic models for the solidification of a pure substance are presented. The first model is based on the two-domain approach, resulting in 2 partial differential equations (PDEs) and one ordinary differential equation (ODE) with 2 boundary conditions, 2 interface conditions, and one initial condition: the Stefan problem.In the second model, the metal column is considered as one-domain, and one PDE is valid for the whole domain. The result is one PDE with two boundary conditions.The models are implemented in MATLAB, and the ODE solver ode23s is used for solving the systems of equations. The models are developed in order to simulate and control the dynamic response of the solidification rate. The control scheme is based on a linear PI controller.     

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.     

Stabilisation for cascade of nonlinear ODEs and counter-convecting transport dynamics

We consider stabilisation for a nonlinear ordinary differential equation (ODE) and counter-convecting transport partial differential equations (PDEs) cascaded system in which the transport coefficients depend on the ODE state. Stability analysis of the closed-loop system is based on the infinite-dimensional backstepping transformations and a Lyapunov functional. A predictor control is proposed such that the closed-loop system is globally asymptotically stable. The proposed design method is illustrated by a single-link manipulator.     

基于LS-SVM求解非线性常微分方程组的近似解

提出了一种基于最小二乘支持向量机(LS-SVM)的改进方法求解非线性常微分方程组初值问题的近似解.利用径向基核函数(RBF)可导的特点对LS-SVM模型进行改进,将含核函数导数形式的LS-SVM模型转化为优化问题进行求解.方法可在原始对偶集中获得近似解的最佳表示,所得近似解连续可微,且精度较高.给出数值算例,通过与真实解的对比验证了所提方法的准确性和有效性.     

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.     

A new symbolic method for solving linear two-point boundary value problems on the level of operators

We present a new method for solving regular boundary value problems for linear ordinary differential equations with constant coefficients (the case of variable coefficients can be adopted readily but is not treated here). Our approach works directly on the level of operators and does not transform the problem to a functional setting for determining the Green’s function.We proceed by representing operators as noncommutative polynomials, using as indeterminates basic operators like differentiation, integration, and boundary evaluation. The crucial step for solving the boundary value problem is to understand the desired Green’s operator as an oblique Moore–Penrose inverse. The resulting equations are then solved for that operator by using a suitable noncommutative Gröbner basis that reflects the essential interactions between basic operators.We have implemented our method as a Mathematica™ package, embedded in the TH$\exists$OREM$\forall$ system developed in the group of Prof. Bruno Buchberger. We show some computations performed by this package.     

Semi-discrete method for generalized Schrödinger-type equations

In this article, a semi-discrete method for solving a class of generalized Schrödinger-type equations is presented. By discretization of the spatial variables, the initial-boundary value problem for partial differential equations can be reduced to the initial value problem for ordinary differential systems. And it is very convenient for numerical analyses and computations.     

Artificial neural networks for solving ordinary and partialdifferential equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE), to systems of coupled ODE and also to partial differential equations (PDE). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galerkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.     

Serre’s reduction of linear partial differential systems with holonomic adjoints

Given a linear functional system (e.g., an ordinary/partial differential system, a differential time-delay system, a difference system), Serre’s reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre’s reduction of underdetermined linear systems of partial differential equations with either polynomial, formal power series or locally convergent power series coefficients, and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial differential systems can be defined by means of only one linear partial differential equation. In the case of polynomial coefficients, we give an algorithm to compute the corresponding equation.     

Direct simulation of the infinitesimal dynamics of semi-discrete approximations for convection-diffusion-reaction problems

In this paper a scheme for approximating solutions of convection-diffusion-reaction equations by Markov jump processes is studied. The general principle of the method of lines reduces evolution partial differential equations to semi-discrete approximations consisting of systems of ordinary differential equations. Our approach is to use for this resulting system a stochastic scheme which is essentially a direct simulation of the corresponding infinitesimal dynamics. This implies automatically the time adaptivity and, in one space dimension, stable approximations of diffusion operators on non-uniform grids and the possibility of using moving cells for the transport part, all within the framework of an explicit method. We present several results in one space dimension including free boundary problems, but the general algorithm is simple, flexible and on uniform grids it can be formulated for general evolution partial differential equations in arbitrary space dimensions.     

Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions

A new straightforward approach for solving ordinary and partial second-order boundary value problems with Neumann boundary conditions is introduced in this research. This approach depends mainly on the Adomian decomposition method with a new definition of the differential operator and its inverse, which has been modified for Neumann boundary conditions. The effectiveness of the proposed approach is verified by several linear and nonlinear examples.