Large Deflection of Tip Loaded Beam with Differential Transformation Method

2011 ◽  
Vol 250-253 ◽  
pp. 1232-1235 ◽  
Author(s):  
Yi Xiao
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Large Deflection ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
The Finite Difference Method ◽  
Large Deflection Problem ◽  
Constant Section

This paper deals with large deflection problem of a cantilever beam with a constant section under the action of a transverse tip load. The differential transformation method (DTM) is used to solve the nonlinear differential equation governing the problem. An approach treats trigonometric nonlinearity is used in DTM. The results obtained from DTM are compared with those results obtained by the finite difference method and they agree well.

A Fast Parallel Implicit Method for Solving Black-Scholes Partial Differential Equation

Information ◽  
2020 ◽  
Vol 23 (3) ◽  
pp. 159-192
Author(s):  
Ikuya Uematsu ◽  
◽  
Lei Li ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Linear Equations ◽  
Financial Derivatives ◽  
Efficient Computation ◽  
Partial Differential ◽  
Black Scholes ◽  
The Finite Difference Method

The Option is well known as one of the typical financial derivatives. In order to determine the price of this option, the finite difference method is used, which must be calculated using the Black―Scholes partial differential equation. In this paper, efficient computation is performed for tridiagonal Toeplitz linear equations which is needed when solving Black―Scholes partial differential equation. Let size of discretization with time is n, and size of discretization for property's value is m, we propose a method to find the solution with the required number of parallel steps of 4n log m, and the required number of processors m + log m.

A Study on Buried Pipeline Response to Fault Movement

1994 ◽  
Vol 116 (1) ◽  
pp. 36-41 ◽  
Author(s):  
Y.-J. Chiou ◽  
S.-Y. Chi ◽  
H.-Y. Chang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Large Deflection ◽  
Strike Slip ◽  
Buried Pipeline ◽  
Fault Movement ◽  
Infinite Beam ◽  
Small Deflection ◽  
The Finite Difference Method

This study investigates the buried pipeline response to strike slip fault movement. The large deflection pipe crossing the fault zone is modeled as an elastica, while the remaining portion of small deflection pipe is modeled as a semi-infinite beam on elastic foundation. The finite difference method is applied for the numerical solution and the results agree qualitatively with the earlier works.

Differential Transformation Method

2017 ◽  
pp. 140-172
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Nonlinear Equation ◽  
Nonlinear Differential Equation ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Differential Transformation

In this chapter, a new linearization procedure based on Differential Transformation Method (DTM) will be presented. The procedure begins with solving nonlinear differential equation by DTM. The effectiveness of the procedure is verified using a heat transfer nonlinear equation. The simulation result shows the significance of the proposed technique.

Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations

1993 ◽  
Vol 60 (1) ◽  
pp. 167-174 ◽  
Author(s):  
N. S. Abhyankar ◽  
E. K. Hall ◽  
S. V. Hanagud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Chaotic Dynamics ◽  
Partial Differential ◽  
Galerkin Approximations ◽  
The Finite Difference Method

The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

ON A NEW DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS

2013 ◽  
Vol 06 (04) ◽  
pp. 1350057 ◽  
Author(s):  
Abdelhalim Ebaid
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Term ◽  
Nonlinear Differential Equations ◽  
Mathematical Structure ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Computational Budget ◽  
Symbolic Formula

The main difficulty in solving nonlinear differential equations by the differential transformation method (DTM) is how to treat complex nonlinear terms. This method can be easily applied to simple nonlinearities, e.g. polynomials, however obstacles exist for treating complex nonlinearities. In the latter case, a technique has been recently proposed to overcome this difficulty, which is based on obtaining a differential equation satisfied by this nonlinear term and then applying the DTM to this obtained differential equation. Accordingly, if a differential equation has n-nonlinear terms, then this technique must be separately repeated for each nonlinear term, i.e. n-times, consequently a system of n-recursive relations is required. This significantly increases the computational budget. We instead propose a general symbolic formula to treat any analytic nonlinearity. The new formula can be easily applied when compared with the only other available technique. We also show that this formula has the same mathematical structure as the Adomian polynomials but with constants instead of variable components. Several nonlinear ordinary differential equations are solved to demonstrate the reliability and efficiency of the improved DTM method, which increases its applicability.

scholarly journals Application of Differential Transformation Method for Solving HIV Model with Anti-Viral Treatment

2020 ◽  
Vol 14 (3) ◽  
pp. 378-388
Author(s):  
Esther Y. Bunga ◽  
Meksianis Z. Ndii
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Kutta Method ◽  
System Of Differential Equations ◽  
Time Step ◽  
Differential Transformation ◽  
Runge Kutta Method ◽  
Runge Kutta

Mathematical models have been widely used to understand complex phenomena. Generally, the model is in the form of system of differential equations. However, when the model becomes complex, analytical solutions are not easily found and hence a numerical approach has been used. A number of numerical schemes such as Euler, Runge-Kutta, and Finite Difference Scheme are generally used. There are also alternative numerical methods that can be used to solve system of differential equations such as the nonstandard finite difference scheme (NSFDS), the Adomian decomposition method (ADM), Variation iteration method (VIM), and the differential transformation method (DTM). In this paper, we apply the differential transformation method (DTM)  to solve system of differential equations. The DTM is semi-analytical numerical technique to solve the system of differential equations and provides an iterative procedure to obtain the power series of the solution in terms of initial value parameters.. In this paper, we present a mathematical model of HIV with antiviral treatment and construct a numerical scheme based on the differential transformation method (DTM) for solving the model. The results are compared to that of Runge-Kutta method. We find a good agreement of the DTM and the Runge-Kutta method for smaller time step but it fails in the large time step.

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