Spatiotemporal Thermal Field Modeling Using Partial Differential Equations With Time-Varying Parameters

2020 ◽  
Vol 17 (2) ◽  
pp. 646-657
Author(s):  
Di Wang ◽  
Kaibo Liu ◽  
Xi Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Thermal Field ◽  
Time Varying ◽  
Partial Differential ◽  
Varying Parameters ◽  
Field Modeling ◽  
Time Varying Parameters

Analytical new soliton wave solutions of the nonlinear conformable time-fractional coupled Whitham–Broer–Kaup equations

2021 ◽  
pp. 2150492
Author(s):  
Delmar Sherriffe ◽  
Diptiranjan Behera ◽  
P. Nagarani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Function Method ◽  
Visual Representations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Varying Parameters ◽  
Wave Solutions ◽  
Soliton Wave Solutions

The study of nonlinear physical and abstract systems is greatly important in order to determine the behavior of the solutions for Fractional Partial Differential Equations (FPDEs). In this paper, we study the analytical wave solutions of the time-fractional coupled Whitham–Broer–Kaup (WBK) equations under the meaning of conformal fractional derivative. These solutions are derived using the modified extended tanh-function method. Accordingly, different new forms of the solutions are obtained. In order to understand its behavior under varying parameters, we give the visual representations of all the solutions. Finally, the graphs are discussed and a conclusion is given.

Modeling and Control of Flexible Transporter System With Arbitrarily Time-Varying Cable Lengths

2003 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ritz Method ◽  
Nonlinear Partial Differential Equations ◽  
Time Varying ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Lyapunov Controller ◽  
And Control

A systematic procedure for deriving the system model of a cable transporter system with arbitrarily time-varying lengths is presented. Two different approaches are used to develop the model, namely, Newton’s Law and Hamilton’s Principle. The derived governing equations are nonlinear partial differential equations. The same results are obtained using the two methods. The Rayleigh-Ritz method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite order partial differential equations into a finite order discretized system. A Lyapunov controller which can both dissipate the vibratory energy and assure the attainment of the desired goal is derived. The validity of the proposed controller is verified by numerical simulation.

A New Exact Approach for Analyzing Free Vibration of SDOF Systems with Nonperiodically Time Varying Parameters

1999 ◽  
Vol 122 (2) ◽  
pp. 175-179 ◽  
Author(s):  
Q. S. Li
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Functional Relation ◽  
Time Varying ◽  
Single Degree Of Freedom ◽  
Varying Parameters ◽  
Exact Approach ◽  
Time Varying Parameters ◽  
Constant Coefficients ◽  
Good Agreement

A new exact approach for analyzing free vibration of single degree of freedom (SDOF) systems with nonperiodically time varying parameters is presented in this paper. The function for describing the variation of mass of a SDOF system with time is an arbitrary continuous real-valued function, and the variation of stiffness with time is expressed as a functional relation with the variation of mass and vice versa. Using appropriate functional transformation, the governing differential equations for free vibration of SDOF systems with nonperiodically time varying parameters are reduced to Bessel’s equations or ordinary differential equations with constant coefficients for several cases, and the corresponding exact analytical solutions are thus obtained. A numerical example shows that the results obtained by the derived exact approach are in good agreement with those calculated by numerical methods, illustrating that the proposed approach is an efficient and exact method. [S0739-3717(00)00902-8]

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