Genelleştirilmiş Rosenau-KdV ve genelleştirilmiş Rosenau-RLW denklemlerinin kollokasyon yöntemi ile nümerik çözümleri

Bu tezde genelleştirilmiş Rosenau-KdV ve genelleştirilmiş Rosenau-RLW denklemlerinin sayısal çözümleri yedinci (septic) dereceden B-spline fonksiyonlar kullanılarak Kollokasyon yöntemi ile elde edilmiştir. Bu tez dört bölümden oluşmaktadır. Tezin birinci bölümünde Sonlu Elemanlar yöntemi, Kollokasyon yöntemi ve B-spline fonksiyonlar hakkında bilgiler sunulmuştur. Tezin ikinci bölümünde geneleştirilmiş Rosenau-KdV denklemi tanıtıldı ve yedinci dereceden B-spline fonksiyonlar kullanılarak Kollokasyon yöntemi ile nümerik çözümleri elde edilmiştir. Tezin üçüncü bölümünde genelleştirilmiş Rosenau-RLW denklemi verilerek yedinci dereceden B-spline fonksiyonlar kullanılarak Kollokasyon yöntemi ile nümerik çözümleri elde edilmiştir. Tezin son bölümünde ise elde ettiğimiz nümerik değerlere ilişkin sonuç ve öneriler sunulmuştur

Galerkin finite element solution for benjamin-bona-mahony-burgers equation with cubic b-splines

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony– Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic Bspline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms L2, L∞ and three invariants I1, I2 and I3 are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation

On the numerical simulation of time-space fractional coupled nonlinear Schrödinger equations utilizing Wendland’s compactly supported function collocation method

This research describes an efficient numerical method based on Wendland’s compactly supported functions to simulate the time-space fractional coupled nonlinear Schrödinger (TSFCNLS) equations. Here, the time and space fractional derivatives are considered in terms of Caputo and Conformable derivatives, respectively. The present numerical discussion is based on the following ways: we first approximate the Caputo fractional derivative of the proposed equation by a scheme order O(∆t2−α), 0 < α < 1 and then the Crank-Nicolson scheme is employed in the mentioned equation to discretize the equations. Second, applying a linear difference scheme to avoid solving nonlinear systems. In this way, we have a linear, suitable calculation scheme. Then, the conformable fractional derivatives of the Wendland’s compactly supported functions are established for the scheme. The stability analysis of the suggested scheme is also examined in a similar way to the classic Von-Neumann technique for the governing equations. The efficiency and accuracy of the present method are verified by solving two examples