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    New Exact Traveling Wave Solutions to the Kawahara Equation using the tanh (?) Expansion Method
    (Springer, 2023) Gasmi, Boubekeur; Moussa, Alaaeddin Amin; Mati, Yazid; Alhakim, Lama Abdulaziz; Akgül, Ali
    Existing approaches to solving the Kawahara equation rely on nonlinear terms with positive powers in auxiliary ordinary differential equations, limiting the number of possible solutions. This paper proposes a tanh (?) expansion method that uses a novel form that accounts for nonlinear terms with both positive and negative powers of tanh (?) to discover six families of exact traveling wave solutions; four of these families are novel, while the other two correspond to solutions previously discovered in the literature. Some solutions with various coefficient values corresponding to the six discovered family solutions are plotted to show how they behave. © 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited.
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    Novel Exact and Solitary Wave Solutions for The Time-Fractional Nonlinear Maccari's System
    (Universal Wiser Publisher, 2023) Gasmi, Boubekeur; Alhakim, Lama; Mati, Yazid; Moussa, Alaaeddin; Akgul, Ali; Wannan, Rania; Asad, Jihad
    The purpose of this research is to find analytical solutions to the time-fractional nonlinear Maccari system. The double auxiliary equation method, which has never been used before, is used to obtain these solutions. The method is cleverly applied, resulting in the generation of nine new exact solitary wave solutions that have never been found before. We also describe the system's dynamic behavior and the bifurcation of traveling waves. Finally, we show some solutions with different coefficient values that correspond to the nine discovered solutions graphically.
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    Solving nonlinear partial differential equations using a novel Cham method
    (Taylor & Francis Ltd, 2023) Gasmi, Boubekeur; Moussa, Alaaeddin; Mati, Yazid; Alhakim, Lama; Akguel, Ali
    Nonlinear partial differential equations (NLPDEs) have been of great interest in recent years due to their numerous applications. While there are several methods for finding exact solutions to various NLPDEs, more solutions are still required. This paper first proposes the Cham method, a new method for solving NLPDEs that can generate eight families of solutions. The method is then successfully employed to solve the (2+1)-dimensional Bogoyavlenskii's breaking soliton equations. The dynamic behaviour of these equations and the bifurcation of traveling waves are also discussed. Finally, we graphically depict some solutions corresponding to some discovered solutions with different coefficient values. The Cham method is general, effective, and adaptable to many NLPDEs.