Partial derivaives
A formal application of the principle of superposition 13 results in a general series solution (sum of all solutions) given by:
∞
𝑛 2 𝜋 2 𝐾 𝐿 2
𝑛𝜋 𝐿
(−
𝑡)
𝑇(𝑥, 𝑡) = ∑ 𝑃 sin (
𝑥)𝑒
𝑛=1
Bibliography
1) Bronson, Richard, and Gabriel B. Costa. (2014) ‘ Schaum's Outlines: Differential Equations. ’ 4th ed. McGraw Hill Education. 2) Dawkins, Paul. Paul's Online Math Notes, (2003). https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivInterp.aspx . Accessed 02/07/2023 3) Wikipedia. ‘ Taylor's Theorem. ’ Wikimedia Foundation. Last modified: 02/07/2023. https://en.wikipedia.org/wiki/Taylor%27s_theorem. Accessed 02/07/2023 4) Dawkins, Paul. Paul's Online Math Notes, (2003). https://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx. Accessed 04/07/2023 5) Dawkins, Paul. Paul's Online Math Notes, (2003). https://tutorial.math.lamar.edu/Classes/CalcIII/HighOrderPartialDerivs.aspx. Accessed 07/07/2023 6) Wikipedia. ‘ Fundamental Theorem of Calculus. ’ Wikimedia Foundation. Last modified: 15/07/2023. https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus. Accessed 09/07/2023 7) Dawkins, Paul. Paul's Online Math Notes, (2003). https://tutorial.math.lamar.edu/Classes/DE/TheHeatEquation.aspx. Accessed 08/07/2023 8) Wikipedia. 2023. ‘ Thermal Conduction. ’ Wikimedia Foundation. Last modified: 31/07/2023. https://en.wikipedia.org/wiki/Thermal_conductivity. Accessed 08/MM/YYYY 9) Oliver, Jim , Ruth Baker, Yves Capdeboscq, Alan Day, Janet Dyson, Peter Howell, and Benjamin Walker. ‘ M5 Fourier Series & PDEs. ’ Oxford University (2021). https://courses.maths.ox.ac.uk/pluginfile.php/25542/mod_resource/content/1/FSPDE- Lectures-Notes.pdf. Accessed 09/07/2023. 10) ‘ Physics Students Need to Know These 5 Methods for Differential Equations ’ . Physics with Elliot. January, 14 2023. Video. https://www.youtube.com/watch?v=0kY3Wpvutfs&t=556s. Accessed 12/MM/YYYY 11) Maths Editor. https://www.mathcha.io/. Accessed 03/07/2023 12) GeoGebra 3D calculator. https://www.geogebra.org/3d?lang=en. Accessed 04/07/2023 13) Desmos Calculator. https://www.desmos.com/calculator . Accessed 04/07/2023
13 Oliver et al (2021), 37-39.
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