Similarity Solution and Heat Transfer Characteristics for a Class of Nonlinear Convection‐Diffusion Equation with Initial Value Conditions

Abstract

A class of nonlinear convection-diffusion equation is studied in this paper. The partial differential equation is converted into nonlinear ordinary differential equation by introducing a similarity transformation. The asymptotic analytical solutions are obtained by using double-parameter transformation perturbation expansion method (DPTPEM). The influences of convection functional coefficient k(z) and power law index n on the heat transport characteristics are discussed and shown graphically. The comparison with the numerical results is presented and it is found to be in excellent agreement. The method and technique used in this paper have the significance in studying other engineering problems.

Use of the repeated integral transformation method to describe the transport of solute in soil

“…In between the SOV method and numerical techniques for solving the transport equation, lies a family of transform methods [ 36 ] to which the similarity transformation method [ 22 , 37 , 38 ] based on group theory [ [39] , [40] , [41] ] belongs. The similarity (Boltzmann) transformation method involves changing the function space into a single ξ spatial variable [ 3 , [42] , [43] , [44] ].…”

“…The similarity (Boltzmann) transformation method involves changing the function space into a single ξ spatial variable [ 3 , [42] , [43] , [44] ]. This method successfully solves several but one-dimensional transport equations [ 3 , 38 , 45 ]. Another drawback of this method is identifying a suitable variable transformation [ 42 ].…”

Fractional View Analysis of Kuramoto–Sivashinsky Equations with Non-Singular Kernel Operators

“…Fractional differential equations are applied to model a wide range of physical problems, including signal processing [11], electrodynamics [12], fluid and continuum mechanics [13], chaos theory [14], biological population models [15], finance [16], optics [17] and financial models [18]. Here, in particular, [19] presents a homotopy perturbation technique for nonlinear transport equations, papers [20][21][22][23][24][25][26] give the application of ADM to different transport models, also including fractional and nonlinear cases, works [27][28][29][30][31][32] provide reviews or/and developments of various numerical approaches to transport/advection-diffusion problems, while [33] proposes perturbational approach to construct analytical approximations based on the double-parameter transformation perturbation expansion method. Finally, the review paper [34] contains an exhaustive review of various modern fractional calculus applications.…”

Analysis of Fractional-Order System of One-Dimensional Keller–Segel Equations: A Modified Analytical Method

“…Here, in particular, papers [26][27][28][29][30][31][32][33] address the application of ADM to various fractional transport models, whilst paper [34] discusses some nonstandard definitions of fractional derivatives. Sources [35][36][37][38] contain developments and/or reviews of various numerical approaches to transport problems, while [39] proposes an interesting perturbational approach to construct analytical approximations. Finally, the review paper [40] contains a comprehensive number of modern applications of fractional calculus.…”