Abstract:In this study, new families of analytical exact solutions of the space fractional Korteweg-de Vries (KdV) equation are presented. Here, the fractional derivative is considered in conformable sense. By utilizing the Jacobi elliptic function expansion method, the solutions are obtained in general form containing the hyperbolic, trigonometric, and rational functions. Also, the complex valued solutions are obtained and some solutions of this equation are demonstrated.
The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.
In this paper, the largest set in the literature of space, time and space-time conformable fractional Phi-4 equations is found by utilizing an analytical method based upon the Jacobi elliptic functions. These solutions are obtained in a general form including trigonometric, rational, complex and hyperbolic functions. Some problems are presented to illustrate the practical application of the proposed method and some of the solutions are also demonstrated with the two-dimensional and three-dimensional graphics.
Recently, some authors have established a number of inequalities involving the minimum eigenvalue for the Hadamard product of M-matrices. In this paper, we improve these results and give some new lower bounds on the minimum eigenvalue for the Hadamard product of an M-matrix A and its inverse {A^{-1}}. Finally, it is shown by the numerical examples that our bounds are also better than some previous results.
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