Abstract
In this research, an unrivalled hybrid scheme which involves the coupling of the new Elzaki integral transform (an improved version of Laplace transform) and a modified differential transform called the projected differential transform (PDTM) have been implemented to solve the generalized Burgers-Fisher's equation; which springs up due to the fusion of the Burgers' and the Fisher's equation; describing convective effects, diffusion transport or interaction between reaction mechanisms, traffic flows; and turbulence; consequently finding meaningful applicability in the applied sciences viz: gas dynamics, fluid dynamics, turbulence theory, reaction-diffusion theory, shock-wave formation, traffic flows, financial mathematics, and so on.Using the proposed Elzaki projected differential transform method (EPDTM), a generalized exact solution (Solitary solution) in form of a Taylor multivariate series has been obtained; of which the highly nonlinear terms and derivatives handled by PDTM have been decomposed without expansion, computation of Adomian or He's polynomials, discretization, restriction of parameters, and with less computational work whilst achieving a highly convergent results when compared to other existing analytical/exact methods in the literature, via comparison tables, 3D plots, convergence plots and fluid-like plots. Thus showing the distinction, novelty and huge advantage of the proposed method as an asymptotic alternative, in providing generalized or solitary wave solution to a wider class of differential equations.
Highlights
One of the militating problems faced in the field of computational mathematics, numerical analysis, and applied sciences is the problem of obtaining an exact solution to models, be it linear or nonlinear
Every model developed or formulated needs to be solved so as to obtain a realistic relationship between the variables or parameters of the model (It could be independent or dependent), to investigate or study the slight change in the model when any of these parameters or variables in question is varied, and to presume the long term effect of the model system. All of these models have been built on differential equations of which a larger proportion of these models are built on Nonlinear differential equations (Ordinary or Partial differential equation of the nonlinear type)
An ordinary differential equation (ODE) is an equation which consists of functions of single variables with their total derivatives, while a partial differential equation (PDE) is that which consists of several independent variables and a dependent variable with partial derivatives
Summary
One of the militating problems faced in the field of computational mathematics, numerical analysis, and applied sciences is the problem of obtaining an exact solution to models, be it linear or nonlinear. Every model developed or formulated needs to be solved so as to obtain a realistic relationship between the variables or parameters of the model (It could be independent or dependent), to investigate or study the slight change in the model when any of these parameters or variables in question is varied (either increased or decreased), and to presume the long term effect of the model system All of these models have been built on differential equations of which a larger proportion of these models are built on Nonlinear differential equations (Ordinary or Partial differential equation of the nonlinear type). On the other hand, when more variables including time are involved in a phenomenon, for example the flow of a fluid in a channel is determined by several variables like the temperature of the fluid, viscosity, pressure, the nature of the channel, and so on; the partial differential equation comes to play here
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