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In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K (m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.

We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.

In this paper, the modified Kudryashov method is proposed to solve fractional differential equations, and Jumarie's modified Riemann-Liouville derivative is used to convert nonlinear partial fractional differential equation to nonlinear ordinary differential equations. The modified Kudryashov method is applied to compute an approximation to the solutions of the space-time fractional modified Benjamin-Bona-Mahony equation and the space-time fractional potential Kadomtsev-Petviashvili equation. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, hyperbolic function solutions, and rational solutions. This method is powerful, efficient, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

Abstract. Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches. Introduction. Michael Grossman and Robert Katz have indicated in[1] that infinitely many calculi can be constructed independently. Each of these calculi provide different perspectives for approaching many problems in science and engineering. Additionally, a mathematical problem which is difficult or impossible to solve in one calculus can be easily revealed through another calculus. E.g. the Volterra calculus [2], called after Vito Volterra, was introduced to define the derivative of dimensional functions that could not be done using the derivative in the Newtonian sense. Independently, Grossmann introduced bigeometric calculus [3] 45 years later, which turned out to be identical to Volterra calculus. These works stimulate the idea that it can be useful to generate a new calculus according to the area of study. With respect to this idea, it seems to be evident that multiplicative and Volterra differential calculus can be used more effectively as a mathematical tool instead of ordinary differential calculus for the mathematical representation of many problems in science and engineering that can be easily represented in these calculi. Indeed, problems related to growth rates can be expressed effectively within the framework of multiplicative calculus [4]. Additionally, recent studies [6,7] show the importance of usage of Volterra differential calculus in mathematical modeling.

Nonlinear fractional partial differential equations have been solved with the help of the extended trial equation method. Based on the fractional derivative in the sense of modified Riemann-Liouville derivative and traveling wave transformation, the fractional partial differential equation can be turned into the nonlinear nonfractional ordinary differential equation. For illustrating the reliability of this approach, we apply it to the generalized third order fractional KdV equation and the fractionalKn,nequation according to the complete discrimination system for polynomial method. As a result, some new exact solutions to these nonlinear problems are successfully constructed such as elliptic integral function solutions, Jacobi elliptic function solutions, and soliton solutions.

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