Analytical solutions for the Black-Scholes equation

In this paper, the Black-Sholes equation (BS) has been applied successfully with the Cauchy-Euler method and the method of separation of variables and new analytical solutions have been found. The linear partial differential equation (PDE) transformed to linear ordinary differential equation (ODE) as well. We acquired three types of solutions including hyperbolic, trigonometric and rational solutions. Descriptions of these methods are given and the obtained results reveal that three methods are tools for exploring partial differential models

New structure for exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation via conformable fractional derivative

In this paper new fractional derivative and direct algebraic method are used to construct exact solutions of the nonlinear time fractional Sharma-Tasso-Olver equation. As a result, three families of exact analytical solutions are obtained. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations

Optical solitons in a power-law media with fourth-order dispersion by three integration methods

In this paper, the extended trial equation method, the $\exp (-\Omega (\eta ))$-expansion method and the $\tan (\phi (\eta )/2)$-expansion method are used in examining the analytical solution of the non-linear Schrödinger equation (NLSE) with fourth-order dispersion. The proposed methods are based on the integration method and a wave transformation. The NLSE with fourth-order dispersion is an equation that arises in soliton radiation, soliton communications with dispersion caused by the hindrance in presence of higher order dispersion terms. We successfully get some solutions with the kink structure

New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod

This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method. Keywords: Extended trial equation method, Longitudinal wave equation in a MEE circular rod, Dark solitons, Bright solitons, Solitary wave, Periodic solitary wav

Analytical treatment in optical metamaterials with anti-cubic law of nonlinearity by improved exp(-Omega(eta))-expansion method and extended sinh-Gordon equation expansion method

Here, the improved exp(-Omega(eta))-expansion method and extended sinh-Gordon equation expansion method are being applied on (1+2)-dimensional non-linear Schrodinger equation (NLSE), optical metamaterials, with anti-cubic nonlinearity. Materials like photovoltaic-photorefractive, polymer and organic consists of spatial solitons and optical nonlinearities, which can be identified by seeking help from NLSE with anti-cubic nonlinearity. Abundant exact traveling wave solutions consisting of free parameters are established in terms of bright, dark, singular, kink-singular, and combined dark-bright soliton solutions. Various arbitrary constants obtained in the solutions help us to discuss the graphical behavior of solutions and also grants flexibility to formulate solutions that can be linked with a large variety of physical phenomena. Moreover, graphical representation of solutions are shown vigorously in order to visualize the behavior of the solutions acquired for the equation

Periodic type and periodic cross-kink wave solutions to the (2+1)-dimensional breaking soliton equation arising in fluid dynamics

In this paper, we have acquired the periodic type and cross-kink wave solutions. In this paper, we use the Hirota bilinear method. With the help of the symbolic calculation and applying the used method, we solve the (2+1)-dimensional Breaking Soliton (BS) equation. We obtain some periodic wave and cross-kink wave that have greatly enriched the existing literature on the BS equation. All solutions have been verified back into its corresponding equation with the aid of the Maple package program via the three-dimensional images and density images with the help of Maple, the physical characteristics of these waves are described very well. These will be widely used to describe many interesting physical phenomena in the fields of gas, plasma, optics, acoustics, heat transfer, fluid dynamics, classical mechanics and so on