Functional approximation for inversion of Laplace transforms via polynomial series

A method for finding the inverse of Laplace transforms using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into Taylor polynomials by use of a suitable transformation. In this paper, the cross product of a polynomial series basis vector is derived in terms of Taylor polynomials, and as a result the inverse of the Laplace transform is obtained, using the most commonly used polynomial series such as Legendre, Chebyshev, and Laguerre. Properties of Taylor series are first briefly presented and the required function is given as a Taylor series with unknown coefficients. Each Laplace transform is converted into a set of simultaneous linear algebraic equations that can be solved to evaluate Taylor series coefficients. The inverse Laplace transform using other polynomial series is then obtained by transforming the properties of the Taylor series to other polynomial series. The method is simple and convenient for digital computation. Illustrative examples are also given,     

Matrix operators for numerically stable representation of stifflinear dynamic systems

A new transformation having features similar to the Laplace transform (but numerically oriented) is developed from the Chebyshev polynomials theory. Signals are represented as vectors of Chebyshev coefficients, and linear subsystems as precomputed matrices. The original problem is preprocessed only once to yield matrix invariants for fast recurrent computations. Theoretical implications of the exact digitizing of a tenth-order transfer function and the reduced-order modeling of a stiff system are discussed     

Approximation of Laplace transform of fractional derivatives via Clenshaw–Curtis integration

In this paper, we approximate the Laplace transform of fractional derivatives via Clenshaw–Curtis integration. The idea of applying Chebyshev polynomial to the numerical computation of integrals is extended to Laplace transform of fractional derivatives. The numerical stability of forward recurrence relations is considered, which depends on the asymptotic behaviour of the coefficients. Error estimation for the Laplace approximation of the fractional derivatives is also considered. Finally, from the numerical examples, the method seems to be promising for approximation of the Laplace transform of fractional derivative.     

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.     

Chebyshev series approximation for solving differential–algebraic equations (DAEs)

The numerical solution of differential–algebraic equations (DAEs) using the Chebyshev series approximation is considered in this article. Two different problems are solved using the Chebyshev series approximation and the solutions are compared with the exact solutions. First, we calculate the power series of a given equation system and then transform it into Chebyshev series form, which gives an arbitrary order for solving the DAE numerically.     

Embedded image coding using laplace transform for Turkish letters

In this paper, a different cryptographic method is introduced by using a Power series transform. A new algorithm for cryptography is produced. The extended Laplace transform of the exponential function is used to encode an explicit text. The key is generated by applying the modular arithmetic rules to the coefficients obtained in the transformation. Here, ASCII codes used to hide the mathematically generated keys to strengthen the encryption. Text steganography is used to make it difficult to break the password. The made encryption is reinforced by image steganography. To hide the presence of the cipher text, it is embedded in another open text with a stenography method. Later, this text is buried in an image. For decryption, it is seen that the inverse of the Power series transform can be used for decryption easily. Experimental results are obtained by making a simulation of the proposed method. As a result, it is stated that the proposed method can be used in crypto machines.

     

Laplace transform analysis of the carbon cycle

A Laplace transform representation is used to describe the changes in atmospheric CO₂ in response to emissions. The formalism gives an explicit representation of generic relations that are less clear when model results are presented as numerical integrations with particular parameter values. In particular, the Laplace transform formalism clarifies some issues involved in inversion of ice-core data and analysis of geosequestration. The airborne fraction is expressed as the emission growth rate multiplied by the Laplace transform of the atmospheric response function, evaluated at the growth rate. This representation emphasises that historical data only capture carbon cycle dynamics over a limited range of time-scales. The Laplace transform formalism provides a basis for expressing uncertainties in the response function in terms of the Padé–Laplace transformation used for fitting sums of exponentials.     

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.     

Error analysis of the Chebyshev series solution of linear time invariant systems

The first part of this paper calculates the error between a function and its m-term Chebyshev series expansion. The second part calculates the error between the integral of a function and its m-term Chebyshev series obtained using the usual operational matrix of integration. The final part derives the error between the exact solution of a state equation and the m-term Chebyshev series solution obtained by the Chebyshev polynomial method.     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

基于图拉普拉斯变换和极限学习机的时间序列预测算法

由于时间效率的约束,多元时间序列预测算法往往存在预测准确率不足的问题。对此,提出基于图拉普拉斯变换和极限学习机的时间序列预测算法。基于图拉普拉斯变换对时间序列进行半监督的特征提取,通过散布矩阵将监督特征和无监督特征进行融合。设计在线的极限学习机学习算法,仅需要在线更新网络的输出权重矩阵即可完成神经网络的学习。利用提取的特征在线训练极限学习机,实现对多元时间序列的实时预测。基于多个数据集进行仿真实验,结果表明该算法有效地提高了预测准确率。     

A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

This paper aims to demonstrate the superiority of the discrete Chebyshev polynomials over the classical Chebyshev polynomials for solving time‐delay fractional optimal control problems (TDFOCPs). The discrete Chebyshev polynomials have been introduced and their properties are investigated thoroughly. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by these polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algabric system. A comparison has been made between the required CPU time and accuracy of the discrete and continuous Chebyshev polynomials methods. The obtained numerical results reveal that utilizing discrete Chebyshev polynomials is more efficient and less time‐consuming in comparison to the continuous Chebyshev polynomials.     

The operational matrix of polynomial series transformation

This paper introduces the operational matrix of polynomial series transformation T that may be applied to transform any polynomial series basis vector to the Taylor polynomials. The matrix is determined for most commonly used polynomial series expansions, such as the Chebyshev, the Laguerre, the Legendre and the Hermite. Using the polynomial series transformation matrix, the corresponding operational matrix of integration of a polynomial series, may easily be determined. Finally, it is shown that all the approximate methods using polynomial-based operational matrices of integration may be connected to the Taylor-series method.     

Linear system identification using numerical computation of Laplace transforms

A method for system identification using sampled values of the initial transient step or impulse response is described. A polynomial fit of the sampled values is made using Lagrange interpolation and the Laplace transform of the output observed is determined. Then the coefficients of the numerator and denominator polynomials of the system transfer function are determined by minimizing the square of the difference between the observed and calculated values of the Laplace transform of the output variable at a number of discrete points. This process is considerably simplified by the use of tables of coefficients for the numerical calculation of Laplace transforms.     

Application of fractional order theory of magneto-thermoelasticity to an infinite perfect conducting body with a cylindrical cavity

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media is given. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. The solution is obtained by a direct approach by using the thermoelastic potential function. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

     

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.     

Optimal quadrature evaluation of integrals of rapidly oscillating functions in Lipschitz interpolation class in case of strong oscillation

An accuracy-optimal quadrature formula is derived to evaluate the Fourier transform of compact functions from an interpolation Lipschitz class. The case of strong oscillation of the integrand is considered. The optimality is substantiated based on the boundary function method, namely, constructing the Chebyshev center and Chebyshev radius in the uncertainty domain of the problem solution.     

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.     

System Laplace-transform estimation from sampled data

This paper presents a method for estimating the Laplace transform of a dynamic system, given its input and output in sampled-data form and corrupted by noise. The estimate is made by first estimating the coefficients of the pulse transfer function relating the input and output and then by converting these estimates to estimates of the Laplace-transform coefficients. Whenever Laplace-transform coefficients are estimated from sampled data, certain knowledge about the signals between the sampling instants must be known a priori or be assumed. In the proposed method this knowledge is used explicitly to relate the coefficients of the Laplace transform to those of theztransform. When this knowledge is correct the estimate Laplace-transform coefficients are asymptotically unbiased. As an illustration, the proposed method has been used to estimate the transfer function of a second-order dynamic system. In this example the variances of the estimates have been compared with the Cramer-Rao bound for unbiased estimates.