Bratu’s problem: A novel approach using fixed-point iterations and Green’s functions

Kafri, H. Q.; Khuri, Sami

doi:10.1016/j.cpc.2015.09.006

Cited by 41 publications

(13 citation statements)

References 23 publications

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“…Performance matrices like MAD, TIC, and ENSE are used to test the efficiency of our approach in each simulation while global performance matrices like GMAD, GTIC, GFIT, and GENSE are used to test the global efficiency of our novel approach. We give the mathematical formulation for these performance matrices in equations (16)(17)(18)(19)…”

Section: Statistical Evaluationmentioning

confidence: 99%

“…Problems arising in the fields of oscillation theory, ignition, and electro-analytical chemistry are solved with the help of a coupled Green's function together with an iterative technique. These problems include Bratu differential equations [17]- [19]. All these procedures have their effectiveness, exactness, robustness, and other characteristics like the ease of implementation, requirement of prior knowledge about the problem and weaknesses over one another.…”

Section: Introductionmentioning

confidence: 99%

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A Soft Computing Approach Based on Fractional Order DPSO Algorithm Designed to Solve the Corneal Model for Eye Surgery

et al. 2020

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In this research, a soft computing approach based on a Nature-inspired technique, the Fractional-Order Darwinian Particle Swarm Optimization (FO-DPSO) algorithm, is hybridized with feed-forward artificial neural network (FF-ANN) to suggest and calculate better solutions for non-linear second-order ordinary differential equation (ODE) representing the corneal shape model (CSM). The unknown weights involved in approximate solutions obtained through ANN are tuned with the help of FO-DPSO. To test the robustness of our approach and conditionality of CSM, we have considered several cases of CSM with different aspects of the problem. Solutions obtained by Adam's method are used as a reference point for the sake of comparison. We establish it that FO-DPSO is a suitable technique for tuning the unknown weights involved in the solution designed with ANNs. Our results suggest that the proposed approach is a suitable candidate for solving real-world problems involving differential equations. INDEX TERMS Non-linear differential equations, meta-heuristics, soft computing, corneal shape model, feed-forward artificial neural networks, fractional order Darwinian particle swarm optimization.

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Section: Statistical Evaluationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Soft Computing Approach Based on Fractional Order DPSO Algorithm Designed to Solve the Corneal Model for Eye Surgery

et al. 2020

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“…3 when a ¼ 2. The integer order case was solved using the semi-analytic and numerical techniques (Hassan and El-Tawil, 2011;Hassan and Semary, 2013;Jalilian, 2010;Kafri and Khuri, 2016;Semary and Hassan, 2015) using spline method (Jalilian, 2010), variational iteration method (Semary and Hassan, 2015), homotopy analysis method (Hassan and El-Tawil, 2011;Hassan and Semary, 2013) and other methods.…”

Section: Numerical Examplesmentioning

confidence: 99%

Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule

Semary

Hassan

Radwan

2017

Journal of the Association of Arab Universities for Basic and A

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This paper presents a semi-analytical method for solving fractional differential equations with strong terms like (exp, sin, cos,.. .). An auxiliary parameter is introduced into the well-known Picard's method and so called controlled Picard's method. The proposed approach is based on a combination of controlled Picard's method with Simpson rule. This approach can cover a wider range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for higher order differential equations. The proposed approach can be effectively applied to Bratu's problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. Also, it is tested on other fractional differential equations like nonlinear fractional order Sine-Gordon equation. The results demonstrate reliability, simplicity and efficiency of the approach developed.

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“…In the last two decades, many published papers have focused on solving (1.1) by analytical (e.g., Adomian decomposition method, homotopy analysis method, variational iteration methods, Laplace transform decomposition method or differential transformation method) and numerical (e.g., B-spline method, the finite difference method, weighted residual method, the shooting method, multigrid-based methods, the Sinc-Galerkin method, collocation methods based on B-spline basis functions, Bessel collocation method) methods, see [13], [14], [15], [51], [52], [7], [8], [9], [16], [17], [29], [32], [35],... An extension of Bratu's problem is the following boundary value problem, socalled Liouville-Bratu-Gelfand problem (see [26], [21], [27], [28], [30], [35], [41]) :…”

Section: Introductionmentioning

confidence: 99%

Theoretical and numerical considerations on Bratu-type problems

Petruşel

Rus

Şerban

2021

Stud. Univ. Babes-Bolyai Math.

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"In this paper we present an heuristic introduction to Bratu problem and we give some variants of Bratu's theorem (G. Bratu, Sur les \'equations int\'egrales non lin\'eaires, Bulletin Soc. Math. France, 42(1914), 113-142). Using the positivity of Green's function, the monotone iterations technique and the contraction principle, some generalizations of Bratu's result are also given. Numerical aspects are also considered."

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Bratu’s problem: A novel approach using fixed-point iterations and Green’s functions

Cited by 41 publications

References 23 publications

A Soft Computing Approach Based on Fractional Order DPSO Algorithm Designed to Solve the Corneal Model for Eye Surgery

A Soft Computing Approach Based on Fractional Order DPSO Algorithm Designed to Solve the Corneal Model for Eye Surgery

Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule

Theoretical and numerical considerations on Bratu-type problems

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