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: | , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/3917866811254d4484e07959b4ad170f |
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| Sumario: | 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> |
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