Oxford Calculus: Heat Equation Derivation

Tom Rocks Maths
Tom Rocks Maths
58.4 هزار بار بازدید - 2 سال پیش - University of Oxford mathematician Dr
University of Oxford mathematician Dr Tom Crawford derives the Heat Equation from physical principles. The Heat Equation is one of the first PDEs studied as an undergraduate maths student. Test yourself with some exercises on the Heat Equation with this FREE worksheet in Maple Learn: learn.maplesoft.com/doc/yllfw9gbg7/trm-heat-equati… Investigate separable solutions to the Heat Equation here: learn.maplesoft.com/d/PQOSOULGKPFHEREGBKJSJRHRKPCP… We start with a one-dimensional long and thin cylindrical rod which has heat flowing through it from left to right. We calculate the change in the internal energy for a short section between x=a and a+h (for small h) using Liebniz Rule. This must be equal to the heat flux in minus the heat flux out of the segment. By dividing both sides of the equation by h and taking the limit as h tends to zero we arrive at the first equation relating the temperature to the heat flux. We then use Fourier’s Law which relates the heat flux to the temperature gradient to rewrite the equation in terms of only the temperature. Finally, the extension of the equation to 2D and 3D is discussed. Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: play.google.com/store/apps/details?id=com.maplesof… Apple: apps.apple.com/us/app/maple-companion/id1466659419 Other videos in the Oxford Calculus series can be found here:    • Oxford Calculus   Finding critical points for functions of several variables:    • Oxford Calculus: Finding Critical Poi...   Classifying critical points using the method of the discriminant:    • Oxford Calculus: Classifying 2D Criti...   Partial differentiation explained:    • Oxford Calculus: Partial Differentiat...   Second order linear differential equations:    • Oxford Mathematics Open Day 2021: Dif...   Integrating factors explained:    • Oxford Calculus: Integrating Factors ...   Solving simple PDEs:    • Oxford Calculus: Solving Simple PDEs   Jacobians explained:    • Oxford Calculus: Jacobians Explained   Separation of variables integration technique explained:    • Oxford Calculus: Separation of Variab...   Solving homogeneous first order differential equations:    • Oxford Calculus: Solving Homogeneous ...   Taylor’s Theorem explained with examples and derivation:    • Oxford Calculus: Taylor's Theorem Exp...   Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel:    / @maplesoft   Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. www.facebook.com/tomrocksmaths/ twitter.com/tomrocksmaths www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
2 سال پیش در تاریخ 1401/04/04 منتشر شده است.
58,447 بـار بازدید شده
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