Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel

<abstract><p>We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor $ \omega $ considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.</p></abstract>

Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions

In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient α significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.

Solution of One-dimensional Partial Differential Equation with Higher-Order Derivative by Double Laplace Transform Method

Many problems in natural and engineering sciences such as heat transfer, elasticity, quantum mechanics, water flow, and others are modelled mathematically by partial differential equations. Some of these problems may be linear, nonlinear, homogeneous, non-homogeneous, and order greater or equal one. Finding the theoretical solution to these problems with less cumbersome techniques is an active area of research in the aforementioned field. In this research paper, we have developed a new application of the double Laplace transform method to solve homogeneous and non-homogeneous linear partial differential equations (pdes) with higher-order derivatives (i.e order n where n≥2) in science and engineering. We discussed a brief theory of double Laplace transforms that helped in its application. The main advantage of our method is the reduction of computational effort in finding solution to pdes. Another major benefit of our method is solving problems in the form of (21) directly by transforming to an algebraic equation where the inverse double Laplace transform is implemented for analytical solution, unlike other integral transform methods that would first transform to a system of ODEs before they are solved, is it also very effective in solving linear high-order partial differential equations and yield fast convergence. We present a well-simplified solution for easier comprehension by upcoming researchers.

The Double Laplace Transform Expressed in terms of the Lerch Transcendent

In this manuscript, the authors derive a formula for the double Laplace transform expressed in terms of the Lerch Transcendent. The log term mixes the variables so that the integral is not separable except for special values of k. The method of proof follows the method used by us to evaluate single integrals. This transform is then used to derive definite integrals in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

Exact Analytical Solutions of Linear Dissipative Wave Equations via Laplace Transform Method

A wave phenomena evolved day after day, as various concepts regarding waves appeared with the passage of time. These phenomena are generally modelled mathematically by partial differential equations (PDEs). In this research, we investigate the exact analytical solutions of one and two dimensional linear dissipative wave equations which are modelled by second order PDEs with use of some initial and boundary conditions. We use double Laplace transform (DLT) and triple Laplace transform (TLT) methods to determine these exact analytical solutions. We provide examples with figures to test effectiveness of this scheme of Laplace transform

Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

<abstract><p>In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 &lt; \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.</p></abstract>

Flexural dynamic response of monopile foundations under linear wave loads

An analytical solution for the dynamic response of submerged slender circular cylindrical structures subjected to linear wave loads is presented. A double Laplace transform with respect to temporal and spatial variables is applied both to motion equation and boundary conditions. The dynamic deflection of the beam is obtained by inversion of the Laplace transform. The latter with respect to spatial variable is obtained analytically, while the one concerning the temporal variable is numerically calculated using Durbin numerical scheme. Results in the case of a representative example for a monopile foundation subjected to Airy waves are presented and discussed, and the analytical result is compared against numerical dynamic and static solutions.