The adaptive operational Tau method for systems of ODEs

As the Tau method, like many other numerical methods, has the limitation of using a fixed step size with some high degree (order) of approximation for solving initial value problems over long intervals, we introduce here the adaptive operational Tau method. This limitation is very much problem dependent and in such case the fixed step size application of the Tau method loses the true track of the solution. But when we apply this new adaptive method the true solution is recovered with a reasonable number of steps. To illustrate the effectiveness of this method we apply it to some stiff systems of ordinary differential equations (ODEs). The numerical results confirm the efficiency of the method.     

Numerical solution of transient heat conduction equation with variable thermophysical properties by the Tau method

In this article, we proposed the operational approach to the Tau method for solving linear and nonlinear one‐dimensional transient heat conduction equations with variable thermophysical properties which can involve heat generation term. To solve heat conduction equation, first we recall the Tau method to obtain a matrix form of the governing differential equation. Then boundary and initial conditions are transformed into a matrix form. Finally the resulting systems of linear or nonlinear algebraic equations are given. Afterwards, efficient error estimation is also introduced for this method. Some numerical examples are given to illustrate the efficiency and high accuracy of the proposed method and also results are compared with solutions obtained by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 964–977, 2014     

Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations

In this work, we propose an extension of the algebraic formulation of the Tau method for the numerical solution of the nonlinear Volterra-Hammerstein integral equations. This extension is based on the operational Tau method with arbitrary polynomial basis functions for constructing the algebraic equivalent representation of the problem. This representation is an special semi lower triangular system whose solution gives the components of the vector solution. We will show that the operational Tau matrix representation for the integration of the nonlinear function can be represented by an upper triangular Toeplitz matrix. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results. Our numerical experiments show that the accuracy of the Tau approximate solution is independent of the selection of the basis functions.     

A solution to the Lane–Emden equation in the theory of stellar structure utilizing the Tau method

In this paper, we propose a Tau method for solving the singular Lane–Emden equation—a nonlinear ordinary differential equation on a semi‐infinite interval. We applied collocation, Galerkin, and Tau methods for solving this problem, and according to the results, the solution of Tau method is the most accurate. The operational derivative and product matrices of the modified generalized Laguerre functions are presented. These matrices, in conjunction with the Tau method, are then utilized to reduce the solution of the Lane–Emden equation to that of a system of algebraic equations. We also present a comparison of this work with some well‐known results and show that the present solution is highly accurate. Copyright © 2012 John Wiley & Sons, Ltd.     

Improving the Accuracy of Chebyshev Tau Method for Nonlinear Differential Problems

The spectral properties convergence of the Tau method allow to obtain good approximate solutions for linear differential problems advantageously. However, for nonlinear differential problems the method may produce ill-conditioned matrices issued from the approximations obtained in the iterations from the linearization process. In this work we introduce a procedure to approximate nonlinear terms in the differential equations and a new way to build the corresponding algebraic problem improving the stability of the overall algorithm. Introducing the linearization coefficients of orthogonal polynomials in the Tau method within the iterative process, we can go further in the degree to approximate the solution of the differential problems, avoiding the consequences of ill-conditioning.     

The performance of a Tau preconditioner on systems of ODEs

In this work, we investigate the new preconditioner of the Tau method, introduced by Ghoreishi and Mohammad Hosseini [13], on systems of ODEs. With a different and relatively straightforward view on the Tau formulation of systems of ODEs this preconditioner is reformulated and explained. An error analysis of the method is also addressed. Some numerical comparisons with the standard Chebyshev-Tau method and the preconditioned method of Cabos confirm the overall efficiency of this preconditioner for systems of ODEs as well.     

The L<Superscript>2</Superscript>-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations

The operational Tau method, a well known method for solving functional equations is employed to approximate the solution of nonlinear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. The unique solvability of the linear Tau algebraic system is discussed. In addition, we provide a rigorous convergence analysis for the Legendre Tau method which indicate that the proposed method converges exponentially provided that the data in the given FIDE are smooth. To do so, Sobolev inequality with some properties of Banach algebras are considered. Some numerical results are given to clarify the efficiency of the method.     

Piecewise finite series solution of nonlinear initial value differential problem

In this paper, we apply a piecewise finite series as a hybrid analytical-numerical technique for solving some nonlinear systems of ordinary differential equations. The finite series is generated by using the Adomian decomposition method, which is an analytical method that gives the solution based on a power series and has been successfully used in a wide range of problems in applied mathematics. We study the influence of the step size and the truncation order of the piecewise finite series Adomian (PFSA) method on the accuracy of the solutions when applied to nonlinear ODEs. Numerical comparisons between the PFSA method with different time steps and truncation orders against Runge-Kutta type methods are presented. Based on the numerical results we propose a low value truncation order approach with small time step size. The numerical results show that the PFSA method is accurate and easy to implement with the proposed approach.     

An efficient step size control for continuation methods

In solving a nonlinear equation by the use of a continuation method one of the crucial problems is the choice of the step sizes. We present a model for the total computational cost of a standard numerical continuation process and solve the problem of optimal step size control for this model. Using the theoretical results as a basis, we develop an adaptive step size algorithm for Newton's method. This procedure is computationally inexpensive and it gives quite satisfactory results compared to some other numerical experiments found in the literature.     

Geometric methods for nonlinear many-body quantum systems

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to write a simple proof of the famous HVZ theorem in the repulsive case. In the second step we recall the method of geometric localization in Fock space as proposed by Dereziński and Gérard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree-Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar-Tomasevich approximation, for certain values of the coupling constant.     

Waveform relaxation methods for implicit differential equations

We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying Runge-Kutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initial-value problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.     

Control of chaos in a piecewise smooth nonlinear system

This paper shows the stabilization of the unstable periodic orbit of any given piecewise smooth system with linear and/or nonlinear characteristics. By utilizing the periodicity of the switching action, we construct the Poincaré mapping including all information of the original system. This mapping offers a first step toward extending a novel technique for controlling chaos based on the appropriate state feedback in piecewise smooth nonlinear systems. We also apply this approach to Rayleigh type oscillator described by the piecewise smooth nonlinear systems.     

Controller design for stabilizing bilinear systems with state-based switching

Some nonlinear systems can be approximated by switching bilinear systems. In this paper, we proposed a method to design state-based stabilizing controller for switching bilinear systems. Based on the similarity between switching bilinear systems and switching linear systems, corresponding switching linear systems are obtained for switching bilinear systems by applying state-based feedback control laws. Instead, we consider asymptotically stabilizing the corresponding switching linear system through solving a number of relaxed LMI conditions. Stabilizing controllers for switching bilinear systems can be derived based on the results of the corresponding switching linear systems. The stability of the controller is proved step by step through the decreasing of the multiple Lyapunov functions along the state trajectory. The effectiveness of the method is demonstrated by both a theoretical example and an example of urban traffic network with traffic signals.     

Solving nonlinear simultaneous equations with a generalization of Brent's method

We describe an implementation of a generalization of Brent's method for solving systems of nonlinear equations. Some important features of the algorithm, like step control, discretization of derivatives and stopping criteria, are discussed. In particular we give numerical experiences which show that a stopping criterion proposed by D. Gay is efficient.     

Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

Discretization of autonomous ordinary differential equationsby numerical methods might, for certain step sizes, generatesolution sequences not corresponding to the underlying flow—so-called‘spurious solutions’ or ‘ghost solutions’.In this paper we explain this phenomenon for the case of explicitRunge-Kutta methods by application of bifurcation theory fordiscrete dynamical systems. An important tool in our analysisis the domain of absolute stability, resulting from the applicationof the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inheritedby the difference scheme induced by the numerical method whilethe stability type of these inherited genuine fixed points iscompletely determined by the method's domain of absolute stability.We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the correspondinglimit step size. Moreover, we show in which way the bifurcationsoccurring at the limit step size are connected to the valuesof the stability function on the boundary of the domain of absolutestability, where we pay special attention to bifurcations leadingto spurious solutions. In order to explain a certain kind ofspurious fixed points which are not connected to the set ofgenuine fixed points, we interprete the domain of absolute stabilityas a Mandeibrot set and generalize this approach to nonlinearproblems.     

Stein implicit Runge-Kutta methods with high stage order for large-scale ordinary differential equations

The Runge-Kutta method is one of the most popular implicit methods for the solution of stiff ordinary differential equations. For large problems, the main drawback of such methods is the cost required at each integration step for computing the solution of a nonlinear system of equations. In this paper, we propose to reduce the cost of the computation by transforming the linear systems arising in the application of Newton's method to Stein matrix equations. We propose an iterative projection method onto block Krylov subspaces for solving numerically such Stein matrix equations. Numerical examples are given to illustrate the performance of our proposed method.     

Convergence of the BFGS-SQP Method for Degenerate Problems

In this paper, we propose a BFGS (Broyden–Fletcher–Goldfarb–Shanno)-SQP (sequential quadratic programming) method for nonlinear inequality constrained optimization. At each step, the method generates a direction by solving a quadratic programming subproblem. A good feature of this subproblem is that it is always consistent. Moreover, we propose a practical update formula for the quasi-Newton matrix. Under mild conditions, we prove the global and superlinear convergence of the method. We also present some numerical results.     

The weighting subspaces of the Tau Method and orthogonal collocation

In this paper we characterise the weighting subspaces associated with two approximation techniques for solving ordinary differential equations: the Tau Method [E.L. Ortiz, The Tau Method, SIAM J. Numer. Anal. 6 (1969) 480-92] and the orthogonal collocation method. We show that approximations constructed by means of these two methods are always expressible in terms of a prescribed orthogonal polynomials basis, by projection on a suitably chosen finite dimensional weighting subspace. We show, in particular, that collocation is a special Tau Method with a twisted basis.     

A Simulation-free Nonlinear Model Order-reduction Approach and Comparison Study

In this paper, a new approach to the model order reduction of nonlinear systems is presented. This approach does not need a simulation of the original system, and therefore, it is suitable for large systems. By separating the linear and nonlinear parts of the original nonlinear model, the idea is to consider the nonlinearities of the resulting system as additional inputs. Based on the linear system from the last step, a known order-reduction method can be applied to find the coefficients of the nonlinear and the linear parts of a reduced-order model. Two different methods from linear-order reduction (balancing and truncation and Eitelberg's method with some modification) are used for this purpose, and their advantages and disadvantages are discussed. For comparison with some known methods in order reduction of nonlinear systems, three other methods are discussed briefly. Finally, a technical nonlinear system is reduced, and different methods are compared.     

Partial differential integral equation model for pricing American option under multi state regime switching with jumps

In this paper, we consider a two dimensional partial differential integral equation (PDIE) model for pricing American option. A nonlinear rationality parameter function for two asset problems is introduced to deal with the free boundary. The rationality parameter function is added in the PDIEs used for pricing American option problems under multi-state regime switching with jumps. The resulting two dimensional nonlinear system of PDIE is then numerically solved. Based on real poles rational approximation, a strongly stable highly efficient and reliable method is developed to solve such complicated systems of PIDEs. The method is build in a predictor corrector style which makes it linearly implicit, therefore, avoids solving nonlinear systems of equations at each time step in all regimes. The method is seen to maintain the stability and convergence for large jump sizes and high volatility in each regime. The impact of regime switching on option prices corresponding to different values interest rate, volatility, and rationality parameter is computed, illustrated by graphs and given in the tables. Convergence results in each regime are presented and time evolution graphs are given to show the effectiveness and reliability of the method.