On the closed form solution of Troesch's problem

Roberts, S. M.; Shipman, J. S.

doi:10.1016/0021-9991(76)90026-7

Cited by 89 publications

(43 citation statements)

References 8 publications

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“…It was found that for certain values of the parameter λ, it undergoes successive bifurcations and shows distinctive chaotic regimes. The observed bifurcation behavior has been found and studied by Roberts and Shipman [15]. They have shown diagrams of continuous and discontinuous solutions.…”

Section: Introductionmentioning

confidence: 78%

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B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

Çağlar¹,

Çağlar²,

Özer

2014

Acta Phys. Pol. A

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A B-spline method is presented for solving the Troesch problem. The numerical approximations to the solution are calculated and then their behavior is studied and commenced. The chaotic dynamics exhibited by the solutions of Troesch's problem as they were derived by the decomposition method approximation are examined and an approximate critical value for the parameter λ is introduced also in this study. For the parameter value slightly less than λ ≈ 2.2, the solutions begin to show successive bifurcations, nally entering chaotic regimes at higher λ values. The eectiveness and accuracy of the B-spline method is veried for dierent values of the parameter, below its critical value, where the rst bifurcation occurs.

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Section: Introductionmentioning

confidence: 78%

“…(14) and (15). To avoid initial uctuations we performed the averaging over the last 200 values of 2000 iterations.…”

Section: Chaotic Behaviour Of Troesch's Equation Approximate Solutionsmentioning

confidence: 99%

B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

Çağlar¹,

Çağlar²,

Özer

2014

Acta Phys. Pol. A

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“…The challenge throughout the years for this nonlinear parametric problem consists of finding the solution for large values of λ. Therefore, in this study, we provide the DG solution of the Troesch's problem for λ=1,3,5,10,15,20 and we show the efficiency of the method compared to existing numerical and semi-analytical results [5], [9], [12]. different values of λ.…”

Section: International Journal Of Applied Physics and Mathematics Vomentioning

confidence: 95%

“…Scott [9] presented the invariant embedded method and Scott and Watts [10]- [11] presented a combined superposition and quasi-linearization procedure as other alternatives to solve the problem. In 1976, an outstanding paper by Roberts and Shipman [12] appeared providing a closed form solution to the problem in terms of Jacobian elliptic functions. Snyman [13] implemented the inverse shooting method and has been successful in solving this problem for large value of λ. Snyman's, Roberts' and Scott's results turned to be considered as benchmark solution for recent studies.…”

Section: Introductionmentioning

confidence: 99%

A New Discontinuous Galerkin Method to Solve Highly Sensitive Troesch’s Problem

Temimi¹

2013

IJAPM

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Abstract-In this paper, we propose a new discontinuous Galerkin finite element (DG) method to solve Troesch's problem, which is highly sensitive for large values of the parameter. This twopoint boundary value problem has been heavily studied since 1960, however, only a few papers have provided a reliable solution for high sensitivity. Therefore, we developed the DG method which has been proved its efficiency for many decades to be a new numerical solver. We demonstrate through computational results compared with those computed by other methods, that the discontinuous Galerkin method provides a quite efficient, accurate and reliable solution. Thus, the DG method is an attractive and competitive alternative to other numerical and semi-analytical techniques to solve highly sensitive nonlinear problems.Index Terms-Troesch's problem, discontinuous galerkin method, nonlinear boundary value problem. I. INTRODUCTIONTroesch's problem, which arises in the investigation of the confinement of a plasma column by radiation pressure, was initially introduced and formulated by Weibel

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“…The problem was first described and solved by Weibel [27]. It has become a widely used test problem, and has been solved many times, including in analytical closed form [28] by using a shooting method [29], by using a Laplace transform decomposition technique [30] and most recently by using a modified Homotopy perturbation technique [31]. The differential equation is…”

Section: Troesch's Problem and Its Solutionmentioning

confidence: 99%

Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems

Ananthaswamy¹,

Rajendran²

2012

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Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu's equation, Troesch's problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.

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On the closed form solution of Troesch's problem

Cited by 89 publications

References 8 publications

B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

A New Discontinuous Galerkin Method to Solve Highly Sensitive Troesch’s Problem

Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems

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