On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory

We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn...

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Autores principales:	Huizeng Qin, Youmin Lu
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	differential equation application multiple solutions with symmetry scientific computation Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/3917866811254d4484e07959b4ad170f
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spelling	oai:doaj.org-article:3917866811254d4484e07959b4ad170f2021-11-25T19:07:01ZOn a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory10.3390/sym131121372073-8994https://doaj.org/article/3917866811254d4484e07959b4ad170f2021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2137https://doaj.org/toc/2073-8994We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>0</mn></msub><mo>,</mo><msub><mi>λ</mi><mo></mo></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>λ</mi><mo></mo></msup></semantics></math></inline-formula> such that this problem has a unique solution when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><msub><mi>α</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> and has three solutions when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>></mo><msub><mi>α</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mo></mo></msub><mo><</mo><mi>λ</mi><mo><</mo><msup><mi>λ</mi><mo></mo></msup><mo>.</mo></mrow></semantics></math></inline-formula> The solutions of this problem are always even functions due to its symmetric boundary values and autonomous characteristics. We use numerical computation to show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4.0686722336</mn><mo><</mo><msub><mi>α</mi><mn>0</mn></msub><mo><</mo><mrow><mn>4.0686722344</mn></mrow></mrow></semantics></math></inline-formula>. This result improves the existing result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>0</mn></msub><mo>≈</mo><mn>4.069</mn></mrow></semantics></math></inline-formula> and increases the accuracy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>α</mi><mn>0</mn></msub></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>8</mn></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula> We developed an algorithm that reduces errors and increases efficiency in our computation. The interval of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> for this problem to have three solutions for given values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is also computed with accuracy up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>14</mn></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula>Huizeng QinYoumin LuMDPI AGarticledifferential equationapplicationmultiple solutions with symmetryscientific computationMathematicsQA1-939ENSymmetry, Vol 13, Iss 2137, p 2137 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	differential equation application multiple solutions with symmetry scientific computation Mathematics QA1-939
spellingShingle	differential equation application multiple solutions with symmetry scientific computation Mathematics QA1-939 Huizeng Qin Youmin Lu On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory
description	We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>0</mn></msub><mo>,</mo><msub><mi>λ</mi><mo></mo></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>λ</mi><mo></mo></msup></semantics></math></inline-formula> such that this problem has a unique solution when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><msub><mi>α</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> and has three solutions when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>></mo><msub><mi>α</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mo></mo></msub><mo><</mo><mi>λ</mi><mo><</mo><msup><mi>λ</mi><mo></mo></msup><mo>.</mo></mrow></semantics></math></inline-formula> The solutions of this problem are always even functions due to its symmetric boundary values and autonomous characteristics. We use numerical computation to show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4.0686722336</mn><mo><</mo><msub><mi>α</mi><mn>0</mn></msub><mo><</mo><mrow><mn>4.0686722344</mn></mrow></mrow></semantics></math></inline-formula>. This result improves the existing result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>0</mn></msub><mo>≈</mo><mn>4.069</mn></mrow></semantics></math></inline-formula> and increases the accuracy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>α</mi><mn>0</mn></msub></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>8</mn></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula> We developed an algorithm that reduces errors and increases efficiency in our computation. The interval of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> for this problem to have three solutions for given values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is also computed with accuracy up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>14</mn></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula>
format	article
author	Huizeng Qin Youmin Lu
author_facet	Huizeng Qin Youmin Lu
author_sort	Huizeng Qin
title	On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory
title_short	On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory
title_full	On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory
title_fullStr	On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory
title_full_unstemmed	On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory
title_sort	on a conjecture for the one-dimensional perturbed gelfand problem for the combustion theory
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/3917866811254d4484e07959b4ad170f
work_keys_str_mv	AT huizengqin onaconjecturefortheonedimensionalperturbedgelfandproblemforthecombustiontheory AT youminlu onaconjecturefortheonedimensionalperturbedgelfandproblemforthecombustiontheory
_version_	1718410270826561536

On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory

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