Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation

In this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2 and L∞ are computed. Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis

(1/G)-Açılım metodunu kullanarak Sawada–Kotera denkleminin hiperbolik yürüyen dalga çözümleri

In this study, we obtain hyperbolic traveling wave solutions of the Sawada–Kotera equation (S-K), using (1/G)-expansion methods. Special values are given to the parameters in the solutions obtained and graphs are drawn. These graphs are presented using a computer package program. In this paper, (1/G)expansion method is applied to reach the goals set. (1/G)-expansion method is an effective and powerful method to obtain the traveling wave solutions of nonlinear partial differential equations.Bu çalışmada, (1/G)-açılım metodunu kullanarak Sawada- Kotera denkleminin (S-K) hiperbolik yürüyen dalga çözümleri elde edildi. Elde edilen çözümlerdeki parametrelere özel değerler verilerek, grafikler çizildi. Bu grafikler bilgisayar paket programı kullanılarak sunuldu. Bu makalede, belirlenen hedefe ulaşmak için (1/G)-açılım metodu uygulandı. (1/G)-açılım metodu lineer olmayan kısmi diferansiyel denklemlerin yürüyen dalga çözümlerini elde etmede etkili ve güçlü bir metottur

Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions

Abstract The exact solution of fractional combined Korteweg-de Vries and modified Korteweg-de Vries (KdV–mKdV) equation is obtained by using the (1/G′) $(1/G^{\prime})$ expansion method. To investigate a geometrical surface of the exact solution, we choose γ=1 $\gamma=1$. The collocation method is applied to the fractional combined KdV–mKdV equation with the help of radial basis for 0<γ<1 $0<\gamma<1$. L2 $L_{2}$ and L∞ $L_{\infty}$ error norms are computed with the Mathematica program. Stability is investigated by the Von-Neumann analysis. Instable numerical solutions are obtained as the number of node points increases. It is shown that the reason for this situation is that the exact solution contains some degenerate points in the Lorentz–Minkowski space

Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation

In this article, the Sinh–Gordon function method and sub-equation method are used to construct traveling wave solutions of modified equal width equation. Thanks to the proposed methods, trigonometric soliton, dark soliton, and complex hyperbolic solutions of the considered equation are obtained. Common aspects, differences, advantages, and disadvantages of both analytical methods are discussed. It has been shown that the traveling wave solutions produced by both analytical methods with different base equations have different properties. 2D, 3D, and contour graphics are offered for solutions obtained by choosing appropriate values of the parameters. To evaluate the feasibility and efficacy of these techniques, a nonlinear evolution equation was investigated, and with the help of symbolic calculation, these methods have been shown to be a powerful, reliable, and effective mathematical tool for the solution of nonlinear partial differential equations