In this paper, we will investigate the validity of numerical inverse Laplace transform algorithms to overcome these difficulties. The rapid growth of fractional-order models leads to the emergence of complicated fractional-order differential equations, and brings forward challenges for solving these complicated fractional-order differential equations. Moreover, some variable-order fractional models and distributed-order fractional models were proposed to understand or describe basic nature in a better way. A growing number of fractional-order differential equation based models were provided to describe physical phenomena and complex dynamic systems. Nowadays, fractional calculus has been applied extensively in science, engineering, mathematics, and so on, ,. Fractional calculus, developed from the field of pure mathematics, was increasingly studied in various fields, ,. Leibniz raised the possibility of generalizing the operation of differentiation to non-integer orders in 1695. In our study, Invlap, Gavsteh and improved NILT, which is simply called NILT in this paper, are tested using Laplace transform of simple and complicated fractional-order differential equations.įractional calculus is a part of mathematics dealing with derivatives of arbitrary order, ,. However, there is a lack of assessments for applying numerical inverse Laplace transform algorithms in solving fractional-order differential equations. Furthermore, some efforts have been made to evaluate the performances of these numerical inverse Laplace transform algorithms. The quotient-difference algorithm based NILT method is more numerically stable giving the same results in a practical way. The algorithm was improved using a quotient-difference algorithm in. The NILT method is based on the application of fast Fourier transformation followed by so-called ɛ ‐algorithm to speed up the convergence of infinite complex Fourier series. Gavsteh numerical inversion of Laplace transform algorithm was introduced in, and the NILT fast numerical inversion of Laplace transforms algorithm was provided in. Based on accelerating the convergence of the Fourier series using the trapezoidal rule, Invlap method for numerical inversion of Laplace transform was proposed in. Direct numerical inversion of Laplace transform algorithm, which is based on the trapezoidal approximation of the Bromwich integral, was introduced in. Weeks numerical inversion of Laplace transform algorithm was provided using the Laguerre expansion and bilinear transformations. Many numerical inverse Laplace transform algorithms have been provided to solve the Laplace transform inversion problems. Motivated by taking advantages of numerical inverse Laplace transform algorithms in fractional calculus, we investigate the validity of applying these numerical algorithms in solving fractional-order differential equations. So, the numerical inverse Laplace transform algorithms are often used to calculate the numerical results. For a complicated differential equation, however, it is difficult to analytically calculate the inverse Laplace transformation. The inverse Laplace transformation can be accomplished analytically according to its definition, or by using Laplace transform tables. Inverse Laplace transform is an important but difficult step in the application of Laplace transform technique in solving differential equations. Laplace transform has been considered as a useful tool to solve integer-order or relatively simple fractional-order differential. 13 Bessel Functions of Order Zero and Unity. 4 Sectionally Rational- and Rows of Delta Functions 28 1. Oregon State University Corvallis, Oregon Eastern Michigan University Ypsilanti, Michigan The Authors Contents Part I. Jolan Eross for her tireless effort and patience while typing this manu script. Latin letters denote (unless otherwise stated) real positive parameters and a possible extension to complex values by analytic continuation will often pose no serious problem. Greek letters denote complex parameters within the given range of validity. Special consideration is given to results involving higher functions as integrand and it is believed that a substantial amount of them is presented here for the first time. Erdelyi and Roberts and Kaufmann (see References). Previous publications include the contributions by A. The usef- ness of this kind of information as a tool in various branches of Mathematics is firmly established. This material represents a collection of integrals of the Laplace- and inverse Laplace Transform type.
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