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In this paper, we study the extraction of some analytical solutions to the nonlinear perturbed sine-Gordon equation with the long Josephson junction properties. The model studied was formed to observe the long Josephson junction properties separated by two superconductors. Moreover, it is also used to explain the Josephson effect arising in the highly nonlinear nature of the Josephson junctions. This provides the shunt inductances to realize a Josephson left-handed transmission line. A powerful scheme is used to extract the complex function solutions. These complex results are used to explain deeper properties of Josephson effects in the frame of impedance. Various simulations of solutions obtained in this paper are also reported. [ABSTRACT FROM AUTHOR]

An editorial is presented on mathematical models in the form of numerical solutions, analytical solutions, approximate solutions, and periodic properties. Topics include Sumudu decomposition method (SDM) finding approximate solutions to two-dimensional fractional partial differential equations; and applying an integral transform-based technique for solving a fractional order Volterra-type integro-differential equation (FVIDE).

In this study, we have applied the modified exp(-Ω(ξ))-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 FROM AUTHOR]

In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation and the (2 + 1)-dimensional cubic Klein-Gordon equation by using the generalized Kudryashov method. After we submitted the general properties of the generalized Kudryashov method in Section 2, we applied this method to these problems to obtain some new analytical solutions, such as rational function solutions, exponential function solutions and hyperbolic function solutions in Section 3. Afterwards, we draw two- and three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]