Solving inverse laplace transform,linear and nonlinear state equations using block-pulse functions

A recursive algorithm is developed for solving the inverse Laplace transform, linear and nonlinear state equations using block-pulse functions. The relationships between the solution of the continuous-time state equation using block-pulse functions and that of the equivalent discrete-time state equation using trapezoidal rule are investigated. A complete computer program is presented for solving the differential equations of linear and nonlinear state equations using block-pulse functions.     

Solution of piecewise-linear ordinary differential equations using waveform relaxation and Laplace transforms

Piecewise-linear (PWL) functions are frequently used to describe the nonlinear branch equations of nonlinear devices in LSI circuits. New techniques for the solution of the differential equations describing the behavior of piecewise-linear circuits will be presented. These techniques are based on the waveform relaxation method to decouple the system equations and Laplace transform techniques to solve the decoupled equations. Several desirable features of the resulting algorithm are discussed.     

A multi-step differential transform method and application to non-chaotic or chaotic systems

The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka–Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi-step DTM, classical DTM and the classical Runge–Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.     

Fuzzy Laplace transforms

In this paper we propose a fuzzy Laplace transform and under the strongly generalized differentiability concept, we use it in an analytic solution method for some fuzzy differential equations (FDEs). The related theorems and properties are proved in detail and the method is illustrated by solving some examples.     

Hybrid laplace transform/weighting function scheme for transient multidimensional conservation equations

A hybrid Laplace transform/weighting function scheme is developed for solving time-dependent multidimensional conservation equations. The new method removes the time derivatives from the governing differential equations using the Laplace transform and solves the associated equation with the weighting function scheme. The similarity transform method is used to treat the complex coefficient system of the equations, which allows the simplest form of complex number functions to be obtained, and then to use the partial fractions method or a numerical method to invert the Laplace transform and transform the functions to the physical plane. Three different examples have been analyzed by the present method. The present method solutions are compared in tables with the exact solutions and those obtained by the other numerical methods. It is found that the present method is a reliable and efficient numerical tool.     

A new analytic solution for fractional chaotic dynamical systems using the differential transform method

Nonlinear differential equations with fractional derivatives give general representations of real life phenomena. In this paper, a modification of the differential transform method (DTM) for solving the nonlinear fractional differential equation is introduced for the first time. The new algorithm is simple and gives an accurate solution. Moreover the new solution is continuous and analytic on each subinterval. A fractional Chen system is considered, to demonstrate the efficiency of the algorithm. The results obtained show good agreement with the generalized Adams–Bashforth–Moulton method.     

A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach

A novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence. From the necessary conditions for optimality of hybrid systems, the hybrid optimal control problem is first converted into a two-point boundary value problem (TPBVP) with additional transverse conditions at the switching times. Then we propose a differential transformation algorithm for solving the TPBVP which may have discontinuities in the state and/or control input at the switching times. Using differential transformation, the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations. The numerical solution is obtained in the form of a truncated Taylor series. By taking advantage of the special properties of the linear subsystems and a quadratic cost functional, the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem. We analyse the error of the numerical solution computed by the differential transformation algorithm and some computational aspects are also discussed. The performance of the differential transformation algorithm is demonstrated through illustrative examples. The differential transformation algorithm has been shown to be simple to be implemented and computationally efficient.     

Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping methods, we develop the Laplace transform methods for solving the free-boundary fractional diffusion equations. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace discretizing the temporal variable. The hyperbola contour integral method is exploited to restore the option values. Meanwhile, the coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation by the fast Laplace transform method is guaranteed. The numerical results confirm that the proposed method outperforms the full finite difference methods in regard to the accuracy and complexity.     

Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.     

Theorems on multidimensional laplace transform for solution of boundary value problems

Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of multidimensional Laplace transform. The general theory for obtaining solutions in this technique is developed in this paper by providing theorems on Laplace transform in n dimensions. Examples are presented for each theorem. Once the basic theorems are established it is possible to derive many useful transform pairs in n variables. Use of the above technique is illustrated by solution of an electrostatic potential problem.     

Solutions of a fractional oscillator by using differential transform method

In this paper, we present an efficient algorithm for solving a fractional oscillator using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of a fractional oscillator. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.     

Finite-time stability of a class of nonlinear fractional-order system with the discrete time delay

This paper investigates the finite-time stability problem of a class of nonlinear fractional-order system with the discrete time delay. Employing the Laplace transform, the Mittag-Leffler function and the generalised Gronwall inequality, the new criterions are derived to guarantee the finite-time stability of the system with the fractional-order 0 < α < 1. Further, we propose the sufficient conditions for ensuring the finite-time stability of the system with the fractional-order 1 < α < 2. Finally, based on the modified Adams–Bashforth–Moulton algorithm for solving fractional-order differential equations with the time delay, we carry out the numerical simulations to demonstrate the effectiveness of the proposed results, and calculate the estimated time of the finite-time stability.     

Unsupervised adaptive neural-fuzzy inference system for solving differential equations

There has been a growing interest in combining both neural network and fuzzy system, and as a result, neuro-fuzzy computing techniques have been evolved. ANFIS (adaptive network-based fuzzy inference system) model combined the neural network adaptive capabilities and the fuzzy logic qualitative approach. In this paper, a novel structure of unsupervised ANFIS is presented to solve differential equations. The presented solution of differential equation consists of two parts; the first part satisfies the initial/boundary condition and has no adjustable parameter whereas the second part is an ANFIS which has no effect on initial/boundary conditions and its adjustable parameters are the weights of ANFIS. The algorithm is applied to solve differential equations and the results demonstrate its accuracy and convince us to use ANFIS in solving various differential equations.     

Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems

This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output system. The artificial neural network (ANN) trial solution of ODE is written as sum of two terms, first one satisfies initial/boundary conditions and contains no adjustable parameters. The second part involves a RBNN model containing adjustable parameters. Network has been trained using the initial weights generated by the coefficients of regression fitting. We have used feed-forward neural network and error back propagation algorithm for minimizing error function. Proposed model has been tested for first, second and fourth-order ODEs. We also compare the results of proposed algorithm with the traditional ANN algorithm. The idea may very well be extended to other complicated differential equations.     

On the nature of block-pulse operational matrices: some further results

The paper presents some further results concerning the methods of the block-pulse functions. Based on an earlier paper, it is shown, that the technique of the block-pulse operational matrices can be simplified by considering its discrete nature and connections with the well-known Z transform. This new approach results in simple recursive algebraic equations and can be used successfully for linear and non-linear, time-invariant and time-varying systems. First, the methods of the block-pulse convolution and stretch matrices are replaced with the method of the so-called ‘block-pulse transform’. Next, the same technique gives the recursive algebraic equations for the non-linear differential equations. The form of the recursive equations depends on the choice of the particular type of block-pulse integrating matrices (conventional, generalized and so on). Then, the block-pulse transform is applied for solving varying systems with and without delays. Finally, some new results concerning the block-pulse technique of the inverse Laplace transform for irrational and transcendental transfer functions are given. A new theorem of the equivalence of the two methods is derived and proved. Many illustrative examples show the superiority of the new approach, which compared with the methods of block-pulse or Walsh matrices seems to be very attractive, simple in the construction and easily used in a computer program.     

On laplace and Dini transformations for multidimensional equations with a decomposable principal symbol

Algorithms for solving linear PDEs implemented in modern computer algebra systems are usually limited to equations with two independent variables. In this paper, we propose a generalization of the theory of Laplace transformations to second-order partial differential operators in ?³ (and, generally, ? ⁿ ) with a principal symbol decomposable into the product of two linear (with respect to derivatives) factors. We consider two algorithms of generalized Laplace transformations and describe classes of operators in ?³ to which these algorithms are applicable. We correct a mistake in [8] and show that Dini-type transformations are in fact generalized Laplace transformations for operators with coefficients in a skew (noncommutative) Ore field. Keywords: computer algebra, partial differential equations, algorithms for solution.     

On the numerical solution of a certain class of non-linear cauchy problems arising from integral equation theory

Summary A numerical algorithm for solving a system of non-linear partial differential integral equations is presented. These equations result from applying the method of invariant imbedding to the solution of a certain class ofFredholm integral equations of the second kind [2]. The algorithm is compared with a more standard method of solution and some numerical results presented.     

Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burgers’ equation

A method to solve weakly non-linear partial differential equations with Volterra series is presented in the context of single-input systems. The solution x(z,t) is represented as the output of a z-parameterized Volterra system, where z denotes the space variable, but z could also have a different meaning or be a vector. In place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems, the problem turns into solving linear differential equations. This paper introduces the method on an example: a dissipative Burgers'equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments. The kernels are computed analytically in the Laplace domain. As a new result, writing the Volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics. Furthermore, the ability of the Volterra system to treat other signals constitutes an improvement for the sound synthesis. It allows the simulation for any regime, including attacks and transients. Numerical simulations are presented and their validity are discussed.     

A method and a software package for numerical solution of the systems of nonlinear ordinary integro-differential-algebraic equations

A method, an algorithm and a software package for automatically solving the ordinary nonlinear integro-differential-algebraic equations (IDAEs) of a sufficiently general form are described. The author understands an automatic solution as obtaining a result without carrying out the stages of selecting a method, programming, and program checking. Both initial and boundary value problems for such equations are solved. It is assumed that the complete set of boundary and initial conditions at the beginning of the integration interval are given. By performing differentiation, the system of IDAEs can be modified, in general, into a system of ordinary nonlinear differential equations (IDEs). The problem of finding the solution of the above-mentioned system on the uniform grid on the integration interval is posed in two forms: as solving the system of IDAEs and as solving the appropriate system of IDEs, where the developed program is to be used. In order to reduce the system of IDAEs and the system of IDEs to the systems of ordinary nonlinear algebraic equations, at every stage of the algorithm the integration and differentiation formulas obtained earlier by N.G. Bandurin are used. Systems similar to those test systems of both nonlinear IDAEs and IDEs considered in this investigation are solved by using the computer programs. It is evident that the coincidence of the results for one and the same system of equations in its different forms can serve as good evidence of the correctness of the obtained results.     

EASY-FIT: a software system for data fitting in dynamical systems

EASY-FIT is an interactive software system to identify parameters in explicit model functions, steady-state systems, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time-dependent partial differential equations with or without algebraic equations. Proceeding from given experimental data, i.e. observation times and measurements, the minimum least squares distance of measured data from a fitting criterion is computed, that depends on the solution of the dynamical system. The software system is implemented in form of a Microsoft Access database running under MS-Windows 95/98/NT/2000. The underlying numerical algorithms are coded in Fortran and are executable independently from the interface. Model functions are either interpreted and evaluated symbolically by a Fortran-similar modeling language, that allows in addition automatic differentiation of nonlinear functions, or by user-provided Fortran subroutines. Received December 30, 2000