JADEs Issue #6: The Heat Equation
The following problem for this week focuses on the popular linear parabolic PDE, the Heat Equation.
PROBLEM 1
Definition
with u = f(x) at t = 0 over an indefinite or even infinite domain on x. So, we get the following indefinite general solution:
Source: H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids,?Clarendon Press, Oxford (1986).
Example
with
at t = 0.
We can get the following general solution:
Example 2
Now, instead of the general or indefinite solution above, we can specify the domain x from -10 to 10 to give us a new particular solution for u(x,t):
领英推荐
Next, here is another linear PDE; this one involving only spatial variables (no time t). We find a particular solution u = u(x,y) instead now.
PROBLEM 2 (EXCERPT)
Definition
Example
NOTE: The arbitrary constant C in above is / should be the summation of three arbitrary constants or C_1 + C_2 + C_3. Each C results after each integration. Since this is a triple integral, there are three resulting arbitrary constants.
Source: Anglin, Linear PDEs of Constant Coefficients, Kindle Direct (2022).
ABOUT THE JOURNAL
The Journal of Applied Differential Equations, otherwise known as JADEs is a free Open Access (OA) eJournal (ISSN# pending) that covers both linear and nonlinear partial differential equations (PDEs) and ordinary differential equations (ODEs) and the analytic methods/solutions to these.
While most journals and proceedings take a very theory-driven approach to PDEs and ODEs, this journal will discuss and illustrate problems from a very pragmatic, workshop problem-solution focus for applied/industrial mathematicians, scientists (physicists, chemists, etc.), engineers (mechanical, electrical, maybe some structural), and others.
Please subscribe if you want to see more problems like the ones above as well as other more relevant PDEs and/or ODEs with applications in mechanics (analytical, classical, Newtonian), electromagnetic field/quantum-based wave mechanics, fluid mechanics and beyond. Enjoy!
ABOUT THE EDITOR
Steve Anglin, M.Sc. Ph.D. (h.c.) is an applied, industrial mathematician primarily teaching and solving partial differential equations (PDEs) with applications in the fields of fluid mechanics (mechanical engineering) as well as some electrical engineering and physics. He is a former lecturer of mathematics at Case and Saint Leo Universities. Steve received his Master of Science (M.Sc.) in applied mathematics from Brown University (Ivy League) and his Hon. Doctorate (Ph.D.(h.c.)) in mathematics from Trinity College. Previously, he published or has been credited on 2,000+ technical trade books with Springer Nature and Pearson Education as well as 2 eZines with O'Reilly Media.
For more, visit:?https://www.amazon.com/author/steveanglin
Email:?[email protected]
18y exp in R&D Product Dev.,Thermal System(HVAC and Powertrain cooling), NVH, Vehicle Development,Testing & EV thermal.
1 年Schrodinger wave equation Ψ - wave function, tells velocity and location of electron, E- energy, H-hamiltonian operator.The Hamiltonian operator, H ^ ψ = E ψ , extracts eigenvalue E from eigenfunction ψ, in which ψ represents the state of a system and E its energy.
DrHuang.com, UNSW alumni
2 年your solution is too complicated. mathHand.com can give simple solution
DrHuang.com, UNSW alumni
2 年Steve A., MSc, PhD(hc) there is another solution https://server.drhuang.com/input/?guess=pdsolve%28ds%28y%2Ct%29%3Da*ds%28y%2Cx%2C2%29%29&inp=ds%28y%2Ct%29%3Da*ds%28y%2Cx%2C2%29&lang=null
DrHuang.com, UNSW alumni
2 年there is another solution https://server.drhuang.com/input/?guess=pdsolve%28ds%28y%2Ct%29%3Da*ds%28y%2Cx%2C2%29%29&inp=ds%28y%2Ct%29%3Da*ds%28y%2Cx%2C2%29&lang=null
2,000+ books published
2 年Thanks for the share!