Analysis of a Two-Dimensional Aeroelastic System Using the Differential Transform Method

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
Xuechuan Wang ◽  
Xiaokui Yue ◽  
Honghua Dai ◽  
Jianping Yuan
Keyword(s):  
Dynamical Behavior ◽  
Integrodifferential Equations ◽  
Control Surface ◽  
Two Dimensional ◽  
Aeroelastic System ◽  
Transform Method ◽  
Nonlinear Responses ◽  
Differential Transform ◽  
Taylor Series Expansions ◽  
System Of Integrodifferential Equations

This paper investigates the nonlinear responses of a typical two-dimensional airfoil with control surface freeplay and cubic pitch stiffness in an incompressible flow. The differential transform (DT) method is applied to the aeroelastic system. Due to the nature of this method, it is capable of providing analytical solutions in forms of Taylor series expansions in each subdomain between two adjacent sampling points. The results demonstrate that the DT method can successfully detect nonlinear aeroelastic responses such as limit cycle oscillations (LCOs), chaos, bifurcation, and flutter phenomenon. The accuracy and efficiency of this method are verified by comparing it with the RK (Henon) method. In addition to ordinary differential equations (ODEs), the DT method is also a powerful tool for directly solving integrodifferential equations. In this paper, the original aeroelastic system of integrodifferential equations is handled directly by the DT method. With no approximation or simplification imposed on the integral terms of aerodynamic function, the resulted solutions are closer to representing the real dynamical behavior.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals Analytical and approximate solution of two-dimensional convection-diffusion problems

2020 ◽  
Vol 10 (1) ◽  
pp. 73-77 ◽  
Author(s):  
Hatıra Günerhan
Keyword(s):  
Approximate Solution ◽  
Research Work ◽  
Convergent Series ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Transform Method ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Differential Transform ◽  
Diffusion Problems

In this work, we have used reduced differential transform method (RDTM) to compute an approximate solution of the Two-Dimensional Convection-Diffusion equations (TDCDE). This method provides the solution quickly in the form of a convergent series. Also, by using RDTM the approximate solution of two-dimensional convection-diffusion equation is obtained. Further, we have computed exact solution of non-homogeneous CDE by using the same method. To the best of my knowledge, the research work carried out in the present paper has not been done, and is new. Examples are provided to support our work.

scholarly journals Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein–Gordon Equation by Using the Reduced Differential Transform Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wayinhareg Gashaw Belayeh ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Type Systems ◽  
Analytical Approximation ◽  
Differential Transform Method ◽  
Two Dimensional ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon ◽  
Reduced Differential Transform Method

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear Klein–Gordon equations (NLKGEs) with quadratic and cubic nonlinearities subject to appropriate initial conditions. The proposed technique has the advantage of producing an analytical approximation in a convergent power series form with a reduced number of calculable terms. Two test examples from mathematical physics are discussed to illustrate the validity and efficiency of the method. In addition, numerical solutions of the test examples are presented graphically to show the reliability and accuracy of the method. Also, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

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