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1
Content available Numerical scheme methods for solving nonlinear pseudo-hyperbolic partial differential equations
Abdulazeez Sadeq Taha, Modanli Mahmut, Husien Ahmad Muhamad
Journal of Applied Mathematics and Computational Mechanics
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2022
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Vol. 21, nr 4
5--15
EN
The numerical solutions to the nonlinear pseudo-hyperbolic partial differentia equation with nonlocal conditions are presented in this study. This equation is solved using the homotopy analysis technique (HAM) and the variational iteration method (VIM). Both strategies are compared and contrasted in terms of approximate and accurate solutions. The results show that the HAM technique is more appropriate, effective, and close to the exact solution than the VIM method. Finally, the graphical representations of the obtained results are given.
2
Content available remote A Numerical Algorithm and a Variational Iteration Technique for Solving Higher Order Fuzzy Integro-differential Equations
Narayanamoorthy S., Murugan K.
Fundamenta Informaticae
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2014
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Vol. 133, nr 4
421--431
EN
In this paper, we present an algorithmic form of the variational iteration method (VIM) to handle both linear and nonlinear higher order fuzzy integro-differential equations. Using parametric form of fuzzy numbers to convert higher order fuzzy integro-differential equation to a system of higher order integro-differential equations in crisp case. By using the variational integration method we find the approximate solution of this system and consequently we obtain an approximation for fuzzy solution of the higher order fuzzy integro-differential equations. The numerical results are examined.
3
Content available Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the Variational Iteration Method
Younesian D., Saadatnia Z., Askari H.
Journal of Theoretical and Applied Mechanics
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2012
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Vol. 50 nr 2
639-652
EN
Closed form expressions are obtained for the dynamic response of an elastic beam rested on a nonlinear foundation in this paper. The nonlinear governing equation is solved using the Variational Iteration Method (VIM). An iteration formulation is constructed based on the VIM and the dynamic responses are then obtained. Frequency responses are presented in a closed form and their sensitivity with respect to the initial amplitudes are investigated. A number of numerical simulations are then carried out and performance and validity of the solution procedure is evaluated in the time domain. It is proved that the VIM is quite a reliable and straightforward technique to solve the corresponding set of coupled nonlinear differential equations.
PL
W pracy przedstawiono zamknięte formy rozwiązań opisujących odpowiedź dynamiczną elastycznej belki spoczywającej na nieliniowo sprężystym podłożu. Równanie ruchu rozwiązano przy pomocy iteracyjnej metody wariacyjnej (VIM). Sformułowanie iteracyjne skonstruowane w oparciu o tę metodę pozwoliło na wyznaczenie odpowiedzi dynamicznej belki. Charakterystyki częstościowe przedstawiono w zamkniętej formie i podkreślono ich wrażliwość na wartość amplitud początkowych. Następnie zaprezentowano wyniki kilku symulacji numerycznych w celu oszacowania wydajności i dokładności zastosowanej procedury rozwiązywania równań ruchu w dziedzinie czasu. Wykazano, że metoda VIM jest wystarczająco prosta i wiarygodna w rozwiązywaniu układu sprzężonych, nieliniowych równań różniczkowych.
4
Content available remote Periodic solutions for highly nonlinear oscillation systems
Ghadimi M., Barari A., Kaliji H. D., Domairry G.
Archives of Civil and Mechanical Engineering
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2012
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Vol. 12, no. 3
389--395
EN
In this paper, Frequency-Amplitude Formulation is used to analyze the periodic behavior of tapered beam as well as two complex nonlinear systems. Many engineering structures, such as offshore foundations, oil platform supports, tower structures and moving arms, are modeled as tapered beams. The results obtained are compared with Variational Iteration Method (VIM) and other analytical methods as well as time marching solution. The results given show the effectiveness and accuracy of the proposed techniques.
5
Content available remote Application of Adomian Decomposition Method and Variational Iteration Method for the solution of fractional Basset differential equation
Ray S. S.
International Journal of Applied Mechanics and Engineering
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2009
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Vol. 14, no 4
1163-1167
EN
In this paper, the Adomian decomposition method (ADM) and variational iteration method (VIM) are implemented to obtain an approximate solution for a fractional Basset equation with an arbitrary order […].The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The approximate solution obtained using the VIM is exactly the same and in good agreement to that obtained by using the ADM.
6
Content available remote Application of Adomian Decomposition Method and Variational Iteration Method for the solution of a fractional (arbitrary) order differential equation
Ray S. S., Bera R. K.
International Journal of Applied Mechanics and Engineering
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2009
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Vol. 14, no 4
1169-1173
EN
In this paper, the Adomian decomposition method (ADM) and variational iteration method (VIM) are implemented to obtain an approximate solution to a fractional differential equation with an arbitrary order […]. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions to different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The approximate solution obtained using the VIM is exactly the same and in good agreement as that obtained by using the ADM.
7
Content available remote Solution of the solidification problem by using the variational iteration method
Hetmaniok E., Słota D., Zielonka A.
Archives of Foundry Engineering
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2009
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Vol. 9, iss. 4
63-68
EN
The paper presents the approximated solution of the solidification problem, modelled with the aid of the one-phase Stefan problem with the boundary condition of the second kind, by using the variational iteration method. For solving this problem one needs to determine the distribution of temperature in the given domain and the position of the moving interface. The proposed method of solution consists of describing the considered problem with a system of differential equations in a domain with known boundary, and solving the received system with the aid of VIM method. The accuracy of the obtained approximated solution is verified by comparing it with the analytical solution.
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