Abstract:In this study, we have applied the modified expp´Ω pξqq-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical solutions have been verified to the (2 + 1)-dimensional Boussinesq water equation by using Wolfram Mathematica 9. We have constructed the two-and three-dimensional surfaces for all analytical solutions obtained in this paper using the same computer program.

Abstract:In this paper we developed a space discrete version of the homotopy analysis method (DHAM) to find the solutions of linear and nonlinear fractional partial differential equations with time derivative α (0 < α ≤ 1). The DHAM contains the auxiliary parameter , which provides a simple way to guarantee the convergence region of solution series. The efficiency and accuracy of the proposed method is demonstrated by test problems with initial conditions. The results obtained are compared with the exact solutions when α = 1. It is shown they are in good agreement with each other.

In this paper, we introduce a combined form of the discrete Sumudu transform method with the discrete homotopy perturbation method to solve linear and nonlinear partial difference equations. This method is called the discrete homotopy perturbation Sumudu transform method(DHPSTM). The results reveal that the introduced method is very efficient, simple and can be applied to other linear and nonlinear difference equations. The nonlinear terms can be easily handled by use of He's polynomials.

Anahtar kelimeler Ayrık homotopi perturbasyon metodu; Caputo kesirli mertebe türev; Kesirli mertebe ayrık difüzyon denklemi; Kesirli mertebe ayrık Schrödinger denklemi; Kesirli mertebe ayrık Burgers denklemi Öz Bu çalışma, lineer ve lineer olmayan zaman kesirli mertebeli kısmi diferensiyel denklemleri çözmek için ayrık uzak biçimli ayrık homotopi perturbasyon metodunu geliştirmiştir. Kesirli mertebe türevler Caputo anlamında göz önüne alınmıştır. Bu metodun başarısı ve uygulanabilirliği bazı örnek problemler ile gösterilmiştir. Elde edilen sonuçlar kesirli mertebe bir olduğunda, tam çözümler ile iyi bir uyumluluk göstermiştir. Bu çalışmada gösterilen metodun kesirli mertebe hesabındaki benzer problemleri çözmesi beklenmektedir.

In this paper, we obtain some sufficient criteria for the oscillation of the solutions of linear impulsive partial difference equations with continuous variables