New Explicit Solutions to the Fractional-Order Burgers’ Equation
M. Hafiz Uddin, Mohammad Asif Arefin, M. Ali Akbar
Abstract
The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mrow> <a:mrow> <a:msup> <a:mrow> <a:mi>G</a:mi> </a:mrow> <a:mrow> <a:mo>′</a:mo> </a:mrow> </a:msup> </a:mrow> <a:mo>/</a:mo> <a:mi>G</a:mi> </a:mrow> </a:mrow> </a:mfenced> <a:mo>,</a:mo> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mrow> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:mi>G</a:mi> </a:mrow> </a:mrow> </a:mfenced> </a:mrow> </a:mfenced> </a:math> -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M2"> <l:mtext>tanh</l:mtext> </l:math> , <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M3"> <n:mtext>sech</n:mtext> </n:math> , <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M4"> <p:mtext>sinh</p:mtext> </p:math> , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.