APPROXIMATE SOLUTION OF FORNBERG–WHITHAM EQUATION BY MODIFIED HOMOTOPY PERTURBATION METHOD UNDER NON-SINGULAR FRACTIONAL DERIVATIVE

The basic idea of this paper is to investigate the approximate solution to a well-known Fornberg–Whitham equation of arbitrary order. We consider the stated problem under ABC fractional order derivative. The proposed derivative is non-local and contains non-singular kernel of Mittag-Leffler type. With the help of Modified Homotopy Perturbation Method (MHPM), we find approximate solution to the aforesaid equations. The required solution is computed in the form of infinite series. The method needs no discretization or collocation and easy to implement to compute the approximate solution that we intend. We also compare our results with that of the exact solution for the initial four terms approximate solution as well as with that computed by the Laplace decomposition method. We also plot the approximate solution of considered model through surface plots. For numerical illustration, we use Matlab throughout this work.

Solutions of Klein-Gordon Equation by the Laplace Decomposition Method and Modified Laplace Decomposition Method

In this article, the Laplace decomposition method and Modified Laplace decomposition method have been employed to obtain the exact and approximate solutions of the Klein-Gordon equation with the initial profile. An approximate solution obtained by these methods is in good agreement with the exact solution and shows that these approaches can solve linear and nonlinear problems very effectively and are capable to reduce the size of computational work.

Double Laplace Decomposition Method and Finite Difference Method of Time-fractional Schrödinger Pseudoparabolic Partial Differential Equation with Caputo Derivative

In this paper, an initial-boundary value problem for a one-dimensional linear time-dependent fractional Schrödinger pseudoparabolic partial differential equation with Caputo derivative of order α ∈ 0,1 is being considered. Two strong numerical methods are employed to acquire the solutions to the problem. The first method used is the double Laplace decomposition method where closed-form solutions are obtained for any α ∈ 0,1 . As the second method, the implicit finite difference scheme is applied to obtain the approximate solutions. To clarify the performance of these two methods, numerical results are presented. The stability of the problem is also investigated.

Solution of the Black-Scholes Equation: LDM and SDM

The main aim of this paper is to discuss a new way of a non-discretization method for the solution of the Black-Scholes equation. Black-Scholes is a mathematical model based on a partial differential equation. The solution of the model is of utmost importance in financial mathematics to estimate option pricing. Several analytical, numerical, and non-discretization methods are existing in the literature to solve the model. Two decomposition methods namely the Laplace decomposition method (LDM) and Sumudu decomposition method (SDM) are adopted for the present study. The results of the present techniques have closed an agreement with an approximate solution which has been obtained with the help of the Adomian Decomposition Method (ADM).

Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems.

Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems.