On Exact Solutions of some Nonlinear Partial Differential Equations of Integer and Fractional Order

Abstract

One of the major consequences of mathematical modeling is nonlinear partial differential equations (NLPDEs). They can be used to analyze and predict the characteristics of many nonlinear real-life phenomena, such as acoustic waves, heat transfer, wave propagation, plasma fluid flow, and diffusion processes, etc. Exact solutions of these NLPDEs gives us the means required to simulate and predict the relevant nonlinear real-life phenomena. Recently, a class of exact solutions (known as soliton solutions) has gained considerable attention due to the potential in mimicking real-life solitary waves. As these types of waves are a very important part of wave propagation in different media, this attention is justified. In this work, we have considered a number of NLPDEs and nonlinear fractional partial differential equations (NLFPDEs) representing certain real-life problems. We have worked out their exact soliton solutions by employing certain mathematical techniques, such as the Generalized Kudryashov Method, Exponential Rational Function Method, Modified Exponential Rational Function Method, (𝐺′ 𝐺2 )-Expansion Method, Auxiliary Equation Method, Khater method, and Generalized Riccati equation mapping method, etc. We have applied these methods to obtain exact solitary wave solutions to a number of NLPDEs and NLFPDEs, such as, NLPDEs representing the van der Waals normal form for fluidized granular matter, the space-time fractional Klein-Gordon equation, space-time fractional Whitham-Broer-Kaup (WBK) equation, time fractional Hirota-Satsuma Coupled Korteweg-de Vries (HSC KdV) equation, (3+1)-dimensional time fractional KdVZakharov-Kuznetsov (KdV-ZK) equation, space-time fractional Boussinesq equation, space-time fractional (2+1)-dimensional breaking soliton equations, space-time fractional Symmetric Regularized Long Wave (SRLW) equation, time fractional (2+1)-dimensional nonlinear Zoomeron equation, space-time fractional Sharma-Tasso-Olver (STO) equation, time fractional Kaup-Kupershmidt (KK) equation, space-time fractional coupled Burgers equations, space-time fractional Zakharov Kuznetsov Benjamin-Bona-Mahony (ZKBBM) equation, ill-posed Boussinesq equation, Nonlinear Longitudinal Wave (NLW) equation, time fractional Sharma-Tasso-Olver (STO) equation and conformable Caudrey-DoddGibbon (CDG) equation. These introduce us to several types of solitary wave solutions like soliton, singular soliton, kink wave, periodic wave, singular kink wave, multiple-soliton wave, multiple periodic solutions, bell-shaped soliton solutions, bright-dark soliton, nontopological (bright) soliton solutions, topological (dark) soliton solutions, cusp-like singular soliton, hyperbolic, trigonometric, exponential and rational solutions. These methods include the use of certain transformations, which transform the given partial differential equation into an ordinary differential equation. For nonlinear fractional partial differential equations (NLFPDEs), an analogous reduction has been achieved by using fractional complex transformations. Besides these suitable transformations, many other strategies have also been used to get exact solutions to the NLPDEs or NLFPDEs at hand. These include using appropriate balancing principles and computer algebra systems such as MAPLE and MATHEMATICA. We have focused on finding methods which could give us such exact solutions which have not been reported yet. Or, even if they have been reported, we have tried to find a more general form of these solutions. To achieve that goal, besides using the already existing techniques, we have also modified the existing methods to hopefully find more general solutions. After the computation of these exact solutions, we have verified them by plugging them back into their respective differential equations. They are found to satisfy their respective differential equation exactly and their solitary wave behavior is captured with the help of graphical simulation.