ATOMFT: solving ODEs and DAEs using Taylor series

Chang, Yao‐Feng; Corliss, George F.

doi:10.1016/0898-1221(94)00193-6

Cited by 88 publications

(62 citation statements)

References 7 publications

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“…It has been used as a powerful numerical scheme for many problems [11][12][13][14][15][16][17][18][19] including chaotic systems [20][21][22][23]. Many numerical algorithms and codes have been developed based on this method [10][11][12][20][21][22][23][24]. However, the abovementioned finiteness of radius of convergence is a serious problem that hinders the use of this method to wide class of differential equations, in particular the nonlinear ones.…”

Section: Introductionmentioning

confidence: 99%

“…Among the many different methods of solving nonlinear differential equations [3][4][5][6][7][8][9], the power series is the most straightforward and efficient [10]. It has been used as a powerful numerical scheme for many problems [11][12][13][14][15][16][17][18][19] including chaotic systems [20][21][22][23]. Many numerical algorithms and codes have been developed based on this method [10][11][12][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

Khawaja

Al‐Mdallal

2018

International Journal of Differential Equations

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It is known that power series expansion of certain functions such as sech( ) diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of sech( ) that is convergent for all . The convergent series is a sum of the Taylor series of sech( ) and a complementary series that cancels the divergence of the Taylor series for ≥ /2. The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1 + 2 ). A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

Khawaja

Al‐Mdallal

2018

International Journal of Differential Equations

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“…Recognizing that a powerful numerical scheme based on this method is already established [34][35][36][44][45][46][47][48], we nonetheless present a thorough investigation of the error associated with this method with the aim of showing how we can systemically reduce errors to infinitesimal values while having the Central Processing Unit (CPU) time within a reasonable range. We will show robustness and efficiency of the method via a highly demanding fluid flow boundaryvalue problem.…”

Section: Introductionmentioning

confidence: 99%

“…It has been used as a powerful numerical scheme for many problems [35][36][37][38][39][40][41][42][43] including chaotic systems [44][45][46][47]. Many numerical algorithms and codes have been developed based on this method [34][35][36][44][45][46][47][48].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Algorithm for Solving Higher‐Order Nonlinear BVPs with an Application on Fluid Flow over a Shrinking Permeable Infinite Long Cylinder

2018

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We present an efficient iterative power series method for nonlinear boundary-value problems that treats the typical divergence problem and increases arbitrarily the radius of convergence. This method is based on expanding the solution around an iterative initial point. We employ this method to study the unsteady, viscous, and incompressible laminar flow and heat transfer over a shrinking permeable cylinder. More precisely, we solve the unsteady nonlinear Navier-Stokes and energy equations after reducing them to a system of nonlinear boundary-value problems of ordinary differential equations. The present method successfully captures dual solutions for both the flow and heat transfer fields and a unique solution at a specific critical unsteadiness parameter. Comparisons with previous numerical methods and an exact solution verify the validity, accuracy, and efficiency of the present method.

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“…It seems there have been no sophisticated Taylor Solvers designed for PCs since 1994 (ATOMFT [3,4]). We expect an application with advanced interactive visualization to provide a user interface with graphics and controls that can be adapted to specific models and operational tasks and realistic visualization of the modeled processes employing all appropriate faculties of the human perception, achievable with advanced hardware and multimedia.…”

Section: Introductionmentioning

confidence: 99%

Using the Taylor Center to Teach ODEs

Gofen

2012

CODEE

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ATOMFT: solving ODEs and DAEs using Taylor series

Cited by 88 publications

References 7 publications

Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

Convergent Power Series of sech⁡(x) and Solutions to Nonlinear Differential Equations

A Numerical Algorithm for Solving Higher‐Order Nonlinear BVPs with an Application on Fluid Flow over a Shrinking Permeable Infinite Long Cylinder

Using the Taylor Center to Teach ODEs

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