720 research outputs found

    Extended Jacobian Elliptic Function Expansion Method and Its Applications in Mathematical Physics

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    In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to some nonlinear evolution equations which play an important role in mathematical physics

    Abundant Exact Soliton Solutions to the Space-Time Fractional Phi-Four Effective Model for Quantum Effects Through the Modern Scheme

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    The space-time fractional Phi-four (PF) equation is measured as a particular case of the familiar Klein-Fock-Gordon (KFG) model and plentiful quantum effects can be investigated through the PF model’s solutions. In this article, the auxiliary equation method (AEM) is employed to attain the traveling wave solutions and in this purpose, the complex wave transformation and Maple software are utilized. The constructed wave solutions are the form likely, hyperbolic, exponential, rational, and trigonometric functions as well as their integration. The physical significance of the obtained solutions for the specific values of the integrated parameters in the course of representing graphs and understood the physical phenomena. It is shown that the AEM is powerful, effective and simple and provide more general traveling wave solutions to the NLEEs

    Efficient high-order finite difference methods for nonlinear Klein–Gordon equations. I: Variants of the phi-four model and the form-I of the nonlinear Klein–Gordon equation

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    In this paper, the problem of solving some nonlinear Klein–Gordon equations (KGEs) is considered. Here, we derive different fourth- and sixth-order explicit and implicit algorithms to solve the phi-four equation and the form-I of the nonlinear Klein–Gordon equation. Stability and consistency of the proposed schemes are studied under certain conditions. Numerical results are presented and then compared with others obtained from some methods already existing in the scientific literature to explain the efficiency of the new algorithms. It is also shown that similar schemes can be proposed to solve many classes of nonlinear KGEs

    Traveling Wave Solutions of ZK-BBM Equation Sine-Cosine Method

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    Travelling wave solutions are obtained by using a relatively new technique which is called sine-cosine method for ZK-BBM equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme

    Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions

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    In this work, we address the problem of solving nonlinear general Klein–Gordon equations (nlKGEs). Different fourth- and sixth-order, stable explicit and implicit, finite difference schemes are derived. These new methods can be considered to approximate all type of Klein–Gordon equations (KGEs) including phi-four, forms I, II, and III, sine-Gordon, Liouville, damped Klein–Gordon equations, and many others. These KGEs have a great importance in engineering and theoretical physics.The higher-order methods proposed in this study allow a reduction in the number of nodes, which might also be very interesting when solving multi-dimensional KGEs. We have studied the stability and consistency of the proposed schemes when considering certain smoothness conditions of the solutions. Additionally, both the typical Dirichlet and some nonlocal integral boundary conditions have been studied. Finally, some numerical results are provided to support the theoretical aspects previously considered

    Using He's variational method to seek the traveling wave solution of PHI-four equation

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    Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation

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    We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity

    Peakon, Cuspon, Compacton, and Loop Solutions of a Three-Dimensional 3DKP(3, 2) Equation with Nonlinear Dispersion

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    We study peakon, cuspon, compacton, and loop solutions for the three-dimensional Kadomtsev-Petviashvili equation (3DKP(3,2) equation) with nonlinear dispersion. Based on the method of dynamical systems, the 3DKP(3,2) equation is shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, compacton, and loop solutions. As a result, the conditions under which peakon, cuspon, compacton, and loop solutions appear are also given
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