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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.

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.