Modified Ji-Huan He’s Frequency Formulation for Large-Amplitude Electron Plasma Oscillations
This paper examines oscillations governed by the generic nonlinear differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><msup><mrow><mi>u</m...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Atoms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2218-2004/12/12/68 |
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| Summary: | This paper examines oscillations governed by the generic nonlinear differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>=</mo><mi>ω</mi></mrow><mrow><mi>p</mi><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mfenced separators="|"><mrow><mn>1</mn><mo>−</mo><mi>u</mi><mo>∓</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>2</mn><msup><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msup></mrow></mfrac></mstyle></mrow></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>p</mi><mn>0</mn></mrow></msub><mo>,</mo><mtext> </mtext><mi>β</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi></mrow></semantics></math></inline-formula> are positive constants. The aforementioned differential equation is of particular importance, as it describes electron plasma oscillations influenced by temperature effects and large oscillation amplitudes. Since no analytical solution exists for the oscillation period in terms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>p</mi><mn>0</mn></mrow></msub><mo>,</mo><mtext> </mtext><mi>β</mi><mo>,</mo><mi>γ</mi></mrow></semantics></math></inline-formula> and the oscillation amplitude, accurate approximations are derived. A modified He’s approach is used to account for the non-symmetrical oscillation around the equilibrium position. The motion is divided into two parts: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>≤</mo><mi>u</mi><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>e</mi><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>e</mi><mi>q</mi></mrow></msub><mo><</mo><mi>u</mi><mo>≤</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> are the minimum and maximum values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>e</mi><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula> is its equilibrium value. The time intervals for each part are calculated and summed to find the oscillation period. The proposed method shows remarkable accuracy compared to numerical results. The most significant result of this paper is that He’s approach can be readily extended to strongly non-symmetrical nonlinear oscillations. It is also demonstrated that the same approach can be extended to any case where each segment of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>+</mo><mi>f</mi><mfenced separators="|"><mrow><mi>u</mi></mrow></mfenced><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> (for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>≤</mo><mi>u</mi><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>e</mi><mi>q</mi></mrow></msub></mrow></semantics></math></inline-formula> and for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>e</mi><mi>q</mi></mrow></msub><mo><</mo><mi>u</mi><mo>≤</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>) can be approximated by a fifth-degree polynomial containing only odd powers. |
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| ISSN: | 2218-2004 |