System of 1st order nonlinear PDE's

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SUMMARY

The discussion centers on a system of first-order nonlinear partial differential equations (PDEs) modeling a solid bed cooled by an uprising gas flow. The equations presented are dv/dt + f(t) * dv/dx = h*(u-v) and du/dt = -h*(u-v), with boundary conditions v(x=0,t) = 0 and dv/dx(x=1,t)=0, and initial condition u(x,t=0) = 1. The variables u and v represent temperature functions dependent on spatial variable x and time t, while f(t) denotes a variable gas velocity function and h is a constant heat transfer coefficient. The user seeks assistance in approximating a solution to this nonlinear PDE system.

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  • Understanding of first-order nonlinear partial differential equations
  • Familiarity with boundary and initial conditions in PDEs
  • Knowledge of heat transfer principles and coefficients
  • Experience with numerical methods for solving PDEs
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  • Research numerical methods for solving nonlinear PDEs, such as the method of characteristics
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Researchers, engineers, and students in applied mathematics, particularly those focusing on heat transfer and fluid dynamics, will benefit from this discussion.

raultrigo
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Modelling a solid bed cooled by an uprising gas flow gave the following system of 1st order NONLINEAR PDEs

dv/dt + f(t) * dv/dx = h*(u-v)
du/dt = -h*(u-v)

BC

v(x=0,t) = 0
dv/dx(x=1,t)=0

IC
u(x,t=0) = 1

where:

u, v are temperatures function of (x,t) range [0,1]
f is a variable gas velocity function of (t)
h is a constant heat transfer coeff
x is a length meassure [0,1]
t is time [0,infinite]

I would appreciate any help on this matter, thank you,
 
Last edited:
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Please find attached a similar problem, this is a System of 1st order LINEAR PDEs,

View attachment SYSTEM OF PDE Anzelius.pdf

I have not been able to manipulate this problem in order to approxiamate a solution, thanks,
 

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