scholarly journals Comparing linear and nonlinear differential equations of differential transformation method by other numerical methods

2014 ◽  
Author(s):  
Che Haziqah Che Hussin ◽  
Adem Kilicman ◽  
Arif Mandangan
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Linear And Nonlinear

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.

Prediction of Creep Strain Relaxations in Biomaterials Using Differential Transformation Method

2018 ◽  
Vol 38 ◽  
pp. 11-22
Author(s):  
Olurotimi Adeleye ◽  
Augustine Eloka ◽  
Gbeminiyi Sobamowo
Keyword(s):  
Numerical Methods ◽  
High Resistance ◽  
Linear Models ◽  
Transformation Method ◽  
High Viscosity ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Resistance To Deformation ◽  
Very High ◽  
Nonlinear Nature

The outcome of most implant failures is tragic. There is an increasing need to reduce the rate of implant failure. While there has been a lot of progress regarding this problem, a lot still needs to be done. The behaviour of biomaterials had been represented using linear models. Linear models failed to capture some certain behaviours in materials due to the nonlinear nature of biomaterials. More work has been done in an attempt to represent the deformation of these biomaterials using non-linear models, which realised success to a degree. However, providing accurate solutions to these models became a problem. Here, An efficient approximate analytical method, differential transformation method (DTM) is provided for prediction of biomaterial deformation. The results of the solutions are found to be in excellent agreements with the results of the numerical methods. It was observed that at high viscosity, the material exhibit very high resistance to deformation and as it decreases, the material allows more deformation, for longer periods of time.Keywords:Biomaterials; Viscoelasticity;deformation;DifferentialTransformation Method;

scholarly journals Switching Point Solution of Second-Order Fuzzy Differential Equations Using Differential Transformation Method

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 231 ◽  
Author(s):  
Nadeem Salamat ◽  
Muhammad Mustahsan ◽  
Malik Saad Missen
Keyword(s):  
Differential Equations ◽  
Numerical Technique ◽  
Transformation Method ◽  
Higher Order ◽  
Differential Transformation Method ◽  
Fuzzy Differential Equation ◽  
Differential Transformation ◽  
Point Solution ◽  
Definition Of ◽  
Higher Order Differential Equations

The first-order fuzzy differential equation has two possible solutions depending on the definition of differentiability. The definition of differentiability changes as the product of the function and its first derivative changes its sign. This switching of the derivative’s definition is handled with the application of min, max operators. In this paper, a numerical technique for solving fuzzy initial value problems is extended to solving higher-order fuzzy differential equations. Fuzzy Taylor series is used to develop the fuzzy differential transformation method for solving this problem. This leads to a single solution for higher-order differential equations.

Generalized Differential Transformation Method for Solving system of Non linear Volterra integro-differential equations of fractional order

2018 ◽  
Vol 1 (25) ◽  
pp. 523-550
Author(s):  
Basim N.Abood ◽  
Eman A.Hussain ◽  
Mayada T. Wazi
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Modified Method ◽  
Non Linear ◽  
Generalized Differential

       In this paper,  the technique of modified Generalized  Differential Transformation Method (GDTM)  is used to solve a system of Non linear integro-differential equations with initial conditions. Moreover, a particular example has been discussed in three different cases to show reliability and the performance of the modified   method. The fractional derivative is considered in the Caputo sense .The approximate solutions are calculated in the form of a convergent series, numerical results explain that this approach is trouble-free to put into practice and correct when applied to systems integro-differential equations.

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