This article addresses the implementation of the new generalized (G'∕G)-expansion method to the Caudrey–Dodd–Gibbon (CDG) equation and the Lax equation which are associated with the fifth-order KdV (fKdV) equation. The method works well to derive a variety of standard and functional closed-form wave solutions with distinct physical structures, such as, soliton, kink, periodic soliton, and bell-shaped soliton solutions. The solutions obtained using this method are useful and adequate than other methods. In order to understand the physical aspects and importance of the method, the attained solutions have been simulated graphically. The extracted results definitely establish that the new generalized(G'∕G)-expansion method is an effective mathematical tool to work out new solutions to different types of local nonlinear evolution equations emerging in applied science and engineering, but this method is not effective in solving nonlocal equations.
In this article, we form the exact wave solutions of the Jimbo-Miwa equation and the Calogero-Bogoyavlenskii-Schiff equation by applying the new generalized (G'/G)-expansion method. We explained the new generalized (G'/G)-expansion method to look for more general traveling wave solutions of the above mentioned equations. The traveling wave solutions attained by this method are in terms of hyperbolic, trigonometric and rational functions. The graphical representation of the obtained solutions is kink soliton, singular kink soliton, singular soliton and singular periodic solution. This method is very significant for extracting exact solutions of NLEEs which habitually occur in mathematical physics, engineering sciences and applied mathematics.
In this article, we establish the exact wave solutions of the Boussinesq equation and the (2 + 1)-dimensional extended shallow water wave equation by applying the new generalized (G'/G)-expansion method. When the condition of the fluid is such that the horizontal length scale is much greater than the vertical length scale, the shallow water equations are mostly suitable. In Ocean engineering, Boussinesq-type equations are commonly used in computer simulations for the model of water waves in shallow seas and harbors. We explained the new generalized (G'/G)-expansion method to seek further general traveling wave solutions of the above mentioned equations. The traveling wave solutions attained by this method are exposed in terms of hyperbolic, trigonometric and rational functions. The shape of the obtained solutions are bell shaped soliton, kink soliton, singular kink soliton, singular soliton, singular periodic solution and compaction. This method is very influential mathematical tool for extracting exact solutions of NLEEs which frequently arise in mathematical physics, engineering sciences and many scientific real world application fields.
The transportation problem is widely applied in the real world. This problem aims to minimize the total shipment cost from a number of sources to a number of destinations. This paper presents a new method named Dhouib-Matrix-TP1, which generates an initial basic feasible solution based on the standard deviation metric with a very reduced number of simple iterations. A comparative study is carried out in order to verify the performance of the proposed Dhouib-Matrix-TP1 heuristic.