scholarly journals A General Scheme for Solving Systems of Linear First-Order Differential Equations Based on the Differential Transform Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ahmed Hussein Msmali ◽  
A. M. Alotaibi ◽  
M. A. El-Moneam ◽  
Badr S. Badr ◽  
Abdullah Ali H. Ahmadini
Keyword(s):  
Differential Equations ◽  
Taylor Series ◽  
General Scheme ◽  
Differential Transform Method ◽  
Algebraic Equations ◽  
Transform Method ◽  
First Order ◽  
Linear Algebraic Equations ◽  
Differential Transform ◽  
First Order Differential Equations

In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

Free vibration analysis of two parallel beams connected together through variable stiffness elastic layer with elastically restrained ends

2016 ◽  
Vol 20 (3) ◽  
pp. 275-287 ◽  
Author(s):  
Alborz Mirzabeigy ◽  
Reza Madoliat ◽  
Mehdi Vahabi
Keyword(s):  
Differential Equations ◽  
Elastic Layer ◽  
Flexural Rigidity ◽  
Free Vibration Analysis ◽  
Variable Stiffness ◽  
Differential Transform Method ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Transform Method ◽  
Differential Transform

In this study, free transverse vibration of two parallel beams connected together through variable stiffness Winkler-type elastic layer is investigated. Euler–Bernoulli beam hypothesis has been applied and the support is considered to be translational and rotational elastic springs in each ends. Linear and parabolic variation has been considered for connecting layer. The equations of motion have been derived in the form of coupled differential equations with variable coefficients. The differential transform method has been applied to obtain natural frequencies and normalized mode shapes of system. Differential transform method is a semi-analytical approach based on Taylor expansion series which converts differential equations to recursive algebraic equations and does not need domain discretization. The results obtained from differential transform method have been validated with the results reported by well-known references in the case of two parallel beams connected through uniform elastic layer. The effects of variation type and total stiffness of connecting layer, flexural rigidity ratio of beams, and boundary conditions on behavior of system are investigated and discussed in detail.

scholarly journals Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon

The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrödinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.

scholarly journals Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Seyyedeh Roodabeh Moosavi Noori ◽  
Nasir Taghizadeh
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Differential Transform Method ◽  
Hybrid Technique ◽  
Transform Method ◽  
Proportional Delays ◽  
Differential Transform ◽  
Computational Work ◽  
Linear And Nonlinear ◽  
First Time

AbstractIn this study, a hybrid technique for improving the differential transform method (DTM), namely the modified differential transform method (MDTM) expressed as a combination of the differential transform method, Laplace transforms, and the Padé approximant (LPDTM) is employed for the first time to ascertain exact solutions of linear and nonlinear pantograph type of differential and Volterra integro-differential equations (DEs and VIDEs) with proportional delays. The advantage of this method is its simple and trusty procedure, it solves the equations straightforward and directly without requiring large computational work, perturbations or linearization, and enlarges the domain of convergence, and leads to the exact solution. Also, to validate the reliability and efficiency of the method, some examples and numerical results are provided.

scholarly journals A fuzzy solution of nonlinear partial differential equations

2021 ◽  
Vol 5 (1) ◽  
pp. 51-63
Author(s):  
Mawia Osman ◽  
◽  
Zengtai Gong ◽  
Altyeb Mohammed Mustafa ◽  
◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Infinite Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Series Expansions ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Fuzzy Solution

In this paper, the reduced differential transform method (RDTM) is applied to solve fuzzy nonlinear partial differential equations (PDEs). The solutions are considered as infinite series expansions which converge rapidly to the solutions. Some examples are solved to illustrate the proposed method.

Sign in / Sign up

Export Citation Format

Share Document