New Approach to Linear Exact Model Matching for a Class of Nonlinear Systems

In this paper, a new approach to the linear exact model matching problem for a class of nonlinear systems, using static state feedback, is presented. This approach reduces the problem of determining the state feedback control law to that of solving a system of first-order partial differential equations. Based on these equations, two major issues are resolved: the necessary and sufficient conditions for the problem to have a solution and the general analytical expression for the feedback control law. Furthermore, the proposed approach is extended to solve the same problem via static output feedback.     

Input-output linearization with simultaneous decoupling by restricted state feedback

The problem of input-output linearization with simultaneousdecoupling via restricted state feedback for nonsquare nonlinearsystems is investigated in this paper. The problem is treatedon the basis of a new algebraic approach whose main featureis that it reduces the determination of the admissible state-feedbackcontrol laws to the solution of an algebraic and a first-orderpartial differential system of equations. Verifiable necessaryand sufficient conditions of an algebraic nature are establishedfor the solvability of the aforementioned problem, which relyon the solvability of these systems of equations. Moreover,an explicit expression for a special admissible restricted state-feedbackcontroller is analytically derived.     

An algebraic approach toA-stable linear multistep-multiderivative integration formulas

A general algebraic approach and some new results are given pertaining to the synthesis of linearA-stable multistep-multiderivative formulas used for integrating stiff differential equations. This problem is shown to be considerably simplified by associating to each formula a special two-variable function, termed the canonical polynomial. In particular, the canonical polynomial approach allows to solve the approximation problem in closed form and provides an easy-to-check algebraic criterion forA-stability. A lower bound is established for the maximum order of accuracy compatible withA-stability, which turns out to be identical to the absolute maximum in some particular cases. It is finally conjectured that this property holds true in general.     

Solvable Symmetry Structures in Differential Form Applications

We investigate symmetry techniques for expressing various exterior differential forms in terms of simplified coordinate systems. In particular, we give extensions of the Lie symmetry approach to integrating Frobenius integrable distributions based on a solvable structure of symmetries and show how a solvable structure of symmetries may be used to find local coordinates for the Pfaffian problem and Darboux's theorem.     

Multi-level meshless methods based on direct multi-elliptic interpolation

A short overview on the direct multi-elliptic interpolation and the related meshless methods for solving partial differential equations is given. A new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain. An error estimation is also derived. To implement the method, quadtree-based multi-level methods are used. The approach avoids the use of large, dense and ill-conditioned matrices and significantly reduces the computational cost.     

An approximate solution of the MHD Falkner–Skan flow by Hermite functions pseudospectral method

Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner–Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal Hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable.     

Three Ways to Solve for Bond Prices in the Vasicek Model

Three approaches in obtaining the closed-form solution of the Vasicek bond pricing problem are discussed in this exposition. A derivation based solely on the distribution of the short rate process is reviewed. Solving the bond price partial differential equation (PDE) is another method. In this paper, this PDE is derived via a martingale approach and the bond price is determined by integrating ordinary differential equations. The bond pricing problem is further considered within the Heath-Jarrow-Morton (HJM) framework in which the analytic solution follows directly from the short rate dynamics under the forward measure.     

Group theoretic method for unsteady free convection flow of a micropolar fluid along a vertical plate in a thermally stratified medium

The group theoretic approach is applied for solving the problem of unsteady natural convection flow of micropolar fluid along a vertical flat plate in a thermally stratified medium. The application of two-parameter transformation group reduces the number of independent variables in the governing system consisting of partial differential equations and a set of auxiliary conditions from three to only one independent variable, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. Numerical solution of the velocity, microrotation and heat transfer have been obtained. The possible forms of the ambient temperature variation with position and time are derived.     

The construction of operational matrix of fractional derivatives using B-spline functions

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. Here we construct the operational matrix of fractional derivative of order α in the Caputo sense using the linear B-spline functions. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus we can solve directly the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the new technique presented in the current paper.     

Alternate solution to generalized Bernoulli equations via an integrating factor: an exact differential equation approach

Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem through a substitution. The purpose of this note is to present an alternative approach using ‘exact methods’, illustrating that a substitution and linearization of the problem is unnecessary. The ideas may be seen as forming a complimentary and arguably simpler approach to Azevedo and Valentino that have the potential to be assimilated and adapted to pedagogical needs of those learning and teaching exact differential equations in schools, colleges, universities and polytechnics. We illustrate how to apply the ideas through an analysis of the Gompertz equation, which is of interest in biomathematical models of tumour growth.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

Parameter estimation in nonlinear stochastic differential equations

We discuss the problem of parameter estimation in nonlinear stochastic differential equations (SDEs) based on sampled time series. A central message from the theory of integrating SDEs is that there exist in general two time scales, i.e. that of integrating these equations and that of sampling. We argue that therefore, maximum likelihood estimation is computationally extremely expensive. We discuss the relation between maximum likelihood and quasi maximum likelihood estimation. In a simulation study, we compare the quasi maximum likelihood method with an approach for parameter estimation in nonlinear SDEs that disregards the existence of the two time scales.     

Nonsmooth Variational Problems for Calibration of Accelerometer Units

Doklady Mathematics - A new formalization of the accelerometer unit calibration problem is proposed within the framework of a guaranteeing approach to estimation. This problem reduces to an...     

Solving approximately a prediction problem for stochastic jump-diffusion systems

In this paper, a new approach to solving a prediction problem for nonlinear stochastic differential systems with a Poisson component is discussed. In this approach, the prediction problem is reduced to an analysis of stochastic jump-diffusion systems with terminating and branching paths. The prediction problem can be approximately solved by using numerical methods for stochastic differential equations and methods for modeling inhomogeneous Poisson flows.     

Bäcklund transformations for the Jacobi system on an ellipsoid

We consider analogues of auto- and hetero-Bäcklund transformations for the Jacobi system on a threeaxis ellipsoid. Using the results in a Weierstrass paper, where the change of times reduces integrating the equations of motion to inverting the Abel mapping, we construct the differential Abel equations and auto-Bäcklund transformations preserving the Poisson bracket with respect to which the equations of motion written in the Weierstrass form are Hamiltonian. Transforming this bracket to the canonical form, we can construct a new integrable system on the ellipsoid with a Hamiltonian of the natural form and with a fourth-degree integral of motion in momenta.     

Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response

A class of reaction-diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction-diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov-Schmidt method and the Implicit Function Theorem.     

A decomposition approach for the integrated vehicle-crew-roster problem with days-off pattern

The integrated vehicle-crew-roster problem with days-off pattern aims to simultaneously determine minimum cost vehicle and daily crew schedules that cover all timetabled trips and a minimum cost roster covering all daily crew duties according to a pre-defined days-off pattern. This problem is formulated as a new integer linear programming model and is solved by a heuristic approach based on Benders decomposition that iterates between the solution of an integrated vehicle-crew scheduling problem and the solution of a rostering problem. Computational experience with data from two bus companies in Portugal and data from benchmark vehicle scheduling instances shows the ability of the approach for producing a variety of solutions within reasonable computing times as well as the advantages of integrating the three problems.     

Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lamé equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lamé equations (a version of the inverse scattering method).     

Matrix displacement mappings in the numerical solution of functional and nonlinear differential equations with the tau method

The use of matrix displacement mappings reduces most matrix operations required in the construction of an approximate solution of a functional or differential equation by means of Ortiz' formulation of the Tau method to index shifts. The coefficient vector of the approximate solution is defined implicitly by a very sparse system of linear algebraic equations. The contributions of the differential or functional operator, and of the supplementary conditions of the problem (initial, boundary, or multipoint conditions) are treated with a single and versatile procedure of remarkable simplicity, which can be easily implemented in a computer. We give two nontrivial examples on the application of this approach: the first is a nonlinear boundary value problem with a continuous locus of singular points and multiple solutions, where stiffness is present, the second is a functional differential equation arising in analytic number theory. In both cases we obtain results of nigh accuracy.