Exact solutions of the variable coefficient kdV and sg type equations

1998 ◽  
Vol 13 (1) ◽  
pp. 25-30 ◽  
Author(s):  
Liu Xiqiang
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient

scholarly journals Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Fanwei Meng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Coefficient Function ◽  
Fractional Partial Differential Equations ◽  
Auxiliary Equation Method

In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

scholarly journals Exact solutions of time fractional heat-like and wave-like equations with variable coefficients

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 689-693 ◽  
Author(s):  
Sheng Zhang ◽  
Ran Zhu ◽  
Luyao Zhang
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Variable Coefficients ◽  
Separation Method ◽  
Variable Separation ◽  
Science And Engineering ◽  
Special Solutions ◽  
Initial And Boundary Conditions ◽  
Variable Separation Method

In this paper, a variable-coefficient time fractional heat-like and wave-like equation with initial and boundary conditions is solved by the use of variable separation method and the properties of Mittag-Leffler function. As a result, exact solutions are obtained, from which some known special solutions are recovered. It is shown that the variable separation method can also be used to solve some others time fractional heat-like and wave-like equation in science and engineering.

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