An Algorithm for Solving Boundary Value Problems

Deeba, Elias; Khuri, Sami; Xie, Shishen

doi:10.1006/jcph.2000.6452

Cited by 135 publications

(34 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The behaviour of the solution near the singular points has been studied numerically [17][18][19] and theoretically [20][21][22][23]. For both one and two-dimensional cases, the Bratu problem has exactly one fold point, whereas the three-dimensional case has infinitely many fold points.…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

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

Ananthaswamy¹,

Rajendran²

2012

View full text Add to dashboard Cite

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.

show abstract

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

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

Ananthaswamy¹,

Rajendran²

2012

View full text Add to dashboard Cite

show abstract

“…Also due to its use in a large variety of applications, many authors have contributed to the study of such problem. Some applications of Bratu problem are the model of thermal reaction process, the Chandrasekhar model of the expansion of the Universe, chemical reaction theory, nanotechnology and radiative heat transfer (see, [28][29][30][31][32]). …”

Section: Introductionmentioning

confidence: 99%

A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations

2015

View full text Add to dashboard Cite

In this article, a new operational matrix method based on shifted Legendre polynomials is presented and analyzed for obtaining numerical spectral solutions of linear and nonlinear second-order boundary value problems. The method is novel and essentially based on reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations in the expansion coefficients of the sought-for spectral solutions. Linear differential equations are treated by applying the Petrov-Galerkin method, while the nonlinear equations are treated by applying the collocation method. Convergence analysis and some specific illustrative examples include singular, singularly perturbed and Bratu-type equations are considered to ascertain the validity, wide applicability and efficiency of the proposed method. The obtained numerical results are compared favorably with the analytical solutions and are more accurate than those discussed by some other existing techniques in the literature.

show abstract

“…(2) It is an unstable two point boundary value problem (BVP) [1,2] and arises from a system of nonlinear ordinary dierential equations which occur in an investigation of the connement of a plasma column by radiation pressure [3]. Robert and Shipman showed that this equation can be solved for λ < 5 [1].…”

Section: Introductionmentioning

confidence: 99%

“…Many dierent techniques, such as decomposition method [3], homotopy perturbation technique [4], Laplace transform decomposition method [5], differential transform method [6], variational iteration method [7], initial value method [1], Adomian decomposition method [8], validating solver for parametric ordinary dierential equations (ODEs) (VSPODE) [9] have been used to solve the Troesch problem. They have also shown that the numerical results do not converge to sufcient accuracy for λ > 1 [3,4].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Çağlar¹,

Çağlar²,

Özer

2014

Acta Phys. Pol. A

View full text Add to dashboard Cite

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.

show abstract

An Algorithm for Solving Boundary Value Problems

Cited by 135 publications

References 14 publications

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

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

A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations

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

Contact Info

Product

Resources

About