Different numerical methods offer for solving Wave equation with diverse boundary conditions. In these methods given partial (Differential equation: DE) change to a normal DE system using equation separating in terms of time variable method. Pade approximation is used for appreciative calculation of exponential matrix function which is created in response of normal DE. Furthermore, its error and stability are investigated by separating method, parallel algorithm is proposed for solving normal Differential equation and numbers of function in serial and parallel methods are compared.

In this paper, we acquire novel traveling Wave solutions of the generalized seventh-order Korteweg–de Vries equation and the seventh-order Kawahara equation as a special case with physical interest. Primarily, we use the advanced $\exp (-\varphi (\xi ))$-expansion method to find new exact solutions of the first equation, by considering two auxiliary equations. Then, we attain some exact solutions of the seventh-order Kawahara equation by using this method with another auxiliary equation, and also using the modified $(G^{'}/G) $-expansion method, where G satisfies a second-order linear ordinary differential equation. Additionally, utilizing the recent scientific instruments, the 2D, 3D, and contour plots are displayed. The solutions obtained in this paper include bright solitons, dark solitary Wave solutions, and multiple dark solitary Wave solutions. It is shown that these two methods provide an effective mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering.

In this paper, extended trial equation method (ETEM) is applied to find exact solutions of (1+1) dimensional nonlinear Ostrovsky equation. We constitute some exact solutions such as soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of this equation via ETEM. Then, we submit the results obtained by using this method.

The current manuscript is concerned with extracting an analytical approximate periodic solution of a damped cubic nonlinear Klein-Gordon equation. The Riemann-Liouville fractional calculus is utilized to obtain an analytic approximate solution. The Homotopy technique is absorbed in the multiple time-spatial scales. The approved scheme yields a generalization of the Homotopy equation; whereas, two different small parameters are adapted. The first parameter concerns with the temporal perturbation, simultaneously, the second one is accompanied by the spatial one. Therefore, the analytic approximate solution needs the two perturbation expansions. This approach conducts more advantages in handling the classical multiple scales method. Furthermore, the initial conditions are included throughout the multiple scale method to achieve a special solution of the governing equation of motion. The analysis ends up deriving two first-order equations within the extended variables and their actual solution is achieved. The procedure adopted here is very promising and powerful in managing similar numerous nonlinear problems arising in physics and engineering. Furthermore, the linearized stability of the corresponding ordinary Duffing differential equation is analyzed. Additionally, some phase portraits are shown.

In this article we use Wavelet transform to solve scalar elastic Wave equation with real boundary condition in earthquake engineering problems (i.e., soil-structure interaction). Multiresolution property, localization and inherent adaptive features of Wavelet transform make this approach appropriate for solving PDEs, moreover, fast algorithms are developed. Working in the framework of Wavelet, here projection schemes are used, i.e., operators (such as derivatives dmf/dxm) are directly projected in the Wavelet spaces. There, the projected operators are sparse and banded matrixes, moreover they are represented in different levels of resolution. The width of the bands could be reduced by thresholding the coefficients of the matrixes, i.e., at first a predefined threshold is assumed, then the coefficients, whose values are smaller than the threshold value, are replaced by zero, thereby, an inherent adaption is attained, increasing the computational speed. The Wavelet families used here is Daubechies Wavelet, an orthogonal family with compact support property. This kind of Wavelet leads to the most regular approximation with the smallest support of Wavelets. Temporal time integration is done by explicit semi-group method. In this method, by using explicit approach, the obtained results are comparable with common implicit schemes. In this work propagation of SH Waves in a infinite region, infinite media containing a hole (such as tunnels), and interaction of a shear wall and a semi-infinite domain are investigated. To simulations the free surfaces, here, the material with nearly zero mechanical properties are used, i.e., an artificial domain (like air) is modeled. Hence, it is not necessary to simulate the free boundaries. Simulation of such boundaries is a challenge problem in Wavelet-based projection schemes, in common Wavelet theories (first generation Wavelet families) it is needed to modify the Wavelets in the vicinity of the boundaries to simulate Dirichlet and Neumann boundary conditions. The infinite boundaries are simulated via absorbing boundaries. They are introduced explicitly by modification the Wave equation, adding artificial viscosity in the computational domain. The viscosity function is zero in domain and increases slightly while approaches to the infinite boundaries. The results indicate that for a grid with constant points, the proposed method is more stable than the common finite difference scheme. Moreover, the results show that interaction of structure and soil is an important factor, for example, for shear walls with higher density, smaller responses are captured during propagation of Waves in the earth. Also when length of propagating Waves is comparable with structure size, high localized responses in the structure will be occurred, therefore common beam element formulation could not be used.