This paper studies bifurcation solutions of the Camassa-Holm equation by using the local Lyapunov-Schmidt method. The Camassa-Holm equation is studied by reduction to an ODE. We find the key function that corresponds to the functional related to this equation and defined on a new domain. The bifurcation analysis of the key function is investigated by the angular singularities. We find the parametric equation of the bifurcation set (caustic) with its geometric description. Also, the bifurcation spreading of the critical points is found.
In this paper, we use the Yang-Laplace transform on Volterra integrodifferential equations of the second kind within the local fractional integral operators to obtain the nondifferentiable approximate solutions. The iteration procedure is based on local fractional derivative operators. This approach provides us with a convenient way to find solution with less computation as compared with local fractional variational iteration method. Some illustrative examples are discussed. The results show that the methodology is very efficient and simple tool for solving integral equations.
This paper studies a nonlinear wave equation's bifurcation solutions of elastic beams situated on elastic bases with small perturbation by using the local method of Lyapunov-Schmidt. We have found the Key function corresponding to the functional related to this equation. The bifurcation analysis of this function has been investigated by the angular singularities. We have found the parametric equation of the bifurcation set (caustic) with the geometric description of this caustic. Also, the critical points' bifurcation spreading has been found.
In this paper, the local fractional variational iteration method (LFVIM) is used for solving linear and nonlinear Fredholm integral equations of the second kind within local fractional derivative operators. To illustrate the ability and simplicity of the method, some examples are provided. The results reveal that the proposed method is very effective and simple and it leads to the exact solution.
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