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The article focuses on investigating traveling wave solutions of the Gardner equation in physics using the (m + (1/G0))-expansion method. It discusses the importance of nonlinear partial differential equations in various fields and explores different methods used to examine the Gardner equation, such as Riccati equations and generalized exponential rational function methods.

In fractional calculus, the fractional differential equation is physically and theoretically important. In this article an efficient numerical process has been developed. Numerical solutions of the time fractional fourth order reaction diffusion equation in the sense of Caputo derivative is obtained by using the implicit method, which is a finite difference method and is developed by increasing the number of iterations. The advantage of the implicit difference scheme is unconditionally stable. The stability analysis and convergency have been proven. A numerical example has been presented, and the validity of the method is supported by tables and graphics. [ABSTRACT FROM AUTHOR]

In this article one of the fractional partial differential equations was solved by finite difference scheme based on five point and three point central space method with discretization in time. We use between the Caputo and the Riemann-Liouville derivative definition and the Grünwald-Letnikov operator for the fractional calculus. The stability analysis of this scheme is examined by using von-Neumann method. A comparison between exact solutions and numerical solutions is made. Some figures and tables are included. [ABSTRACT FROM AUTHOR]