Travelling Wave Solutions of the Non-Linear Wave Equations

Haider, Jamil Abbas; Gul, Sana; Rahman, Jamshaid Ul; Zaman, F. D.

doi:10.2478/ama-2023-0027

Cited by 10 publications

(2 citation statements)

References 39 publications

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“… Our solutions and are likely similar to the solutions mentioned in Eqs. ( 73 ) and ( 81 ) respectively in 57 .…”

Section: Discussion Of the Resultsmentioning

confidence: 99%

Exact solutions of the (3+1)-generalized fractional nonlinear wave equation with gas bubbles

Seadawy,

Ali,

Altalbe

et al. 2024

Sci Rep

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In this manuscript, we implement the travelling wave solutions of the fractional (3+1) generalized computational nonlinear wave equation with gas bubbles via application of five mathematical methods. Liquids with gas bubbles primarily arise in various applications like science, engineering, and mathematical physics. The obtained solitary waves solutions have fruitful applications in engineering, science, life, nature and physics. Several novel soliton solutions of concerned model are established in the form of hyperbolic, trigonometric, exponential and rational functions. To handle all calculations and verification of obtained results, computational software Mathematica 12.1 is used. For the demonstration of the physical behaviour of concern model, some solutions are plotted graphical in 2-dimensional and 3-dimensional by imparting specific values to the parameters under constrain conditions. Finally, we intrigue both two and three dimensional to explain the physical behavior of the model.

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“… Our solutions and are likely similar to the solutions mentioned in Eqs. ( 73 ) and ( 81 ) respectively in 57 .…”

Section: Discussion Of the Resultsmentioning

confidence: 99%

Exact solutions of the (3+1)-generalized fractional nonlinear wave equation with gas bubbles

Seadawy,

Ali,

Altalbe

et al. 2024

Sci Rep

View full text Add to dashboard Cite

show abstract

“…However, this problem has an exact analytic solution within an integral formula [14]. Therefore, getting an exact, bounded approximated solution is helpful [15][16]. A simple pendulum has the most basic and comprehensive structure, which serves as the foundation for numerous complicated applications.…”

Section: Introductionmentioning

confidence: 99%

The Variational Iteration Method for a Pendulum with a Combined Translational and Rotational System

Amir,

Ashraf,

Haider

2024

Acta Mechanica Et Automatica

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The dynamic analysis of complex mechanical systems often requires the application of advanced mathematical techniques. In this study, we present a variation iteration-based solution for a pendulum system coupled with a rolling wheel, forming a combined translational and rotational system. Furthermore, the Lagrange multiplier is calculated using the Elzaki transform. The system under investigation consists of a pendulum attached to a wheel that rolls without slipping on a horizontal surface. The coupled motion of the pendulum and the rolling wheel creates a complex system with both translational and rotational degrees of freedom. To solve the governing equations of motion, we employ the variation iteration method, a powerful numerical technique that combines the advantages of both variational principles and iteration schemes. The Lagrange multiplier plays a crucial role in incorporating the constraints of the system into the equations of motion. In this study, we determine the Lagrange multiplier using the Elzaki transform, which provides an effective means to calculate Lagrange multipliers for constrained mechanical systems. The proposed solution technique is applied to analyse the dynamics of a pendulum with a rolling wheel system. The effects of various system parameters, such as the pendulum length, wheel radius and initial conditions, are investigated to understand their influence on the system dynamics. The results demonstrate the effectiveness of the variation iteration method combined with the Elzaki transform in capturing the complex behaviour of a combined translational and rotational system. The proposed approach serves as a valuable tool for analysing and understanding the dynamics of similar mechanical systems encountered in various engineering applications.

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Exact solutions of Euler–Bernoulli beams

et al. 2023

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In numerous real-world applications, transverse vibrations of beams are nonlinear in nature. It is a task to solve nonlinear beam systems due to their substantial dependence on the 4 variables of the system and the boundary conditions. To comprehend the nonlinear vibration characteristics, it is essential to do a precise parametric analysis. This research demonstrates an approximation solution for odd and even nonlinear transverse vibrating beams using the Laplace-based variation iteration method, and the formulation of the beams depends on the Galerkin approximation. For the solution of the nonlinear differential equation, this method is efficient as compared to the existing methods in the literature because the solutions exactly match with the numerical solutions. The Laplace-based variation iteration method has been used for the first time to obtain the solution to this important problem. To demonstrate the applicability and precision of the Laplace-based iteration method, several initial conditions are applied to the governing equation for nonlinearly vibrating transverse beams. The natural frequencies and periodic response curves are computed using Laplace-based VIM and compared with the Runge–Kutta RK4 method. In contrast to the RK4, the results demonstrate that the proposed method yields excellent consensus. The Lagrange multiplier is widely regarded as one of the most essential concepts in variational theory. The result obtained are displayed in the table form. Highlights The highlights of the solution of the Euler–Bernoulli beam equation with quintic nonlinearity using Lagrange multiplier are: 1. Introducing the constraint of the boundary conditions into the equation using Lagrange multipliers. 2. Formulating the equations for the Lagrange multipliers and the deflection of the beam. 3. Solving the resulting system of algebraic equations using numerical methods. 4. Obtaining the deflection of the beam as a function of its length and the applied load. 5. Analyzing the behavior of the beam under different loads and boundary conditions.

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Travelling Wave Solutions of the Non-Linear Wave Equations

Cited by 10 publications

References 39 publications

Exact solutions of the (3+1)-generalized fractional nonlinear wave equation with gas bubbles

Exact solutions of the (3+1)-generalized fractional nonlinear wave equation with gas bubbles

The Variational Iteration Method for a Pendulum with a Combined Translational and Rotational System

Exact solutions of Euler–Bernoulli beams

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