System of 1st order nonlinear PDE's

In summary, the conversation discussed the system of 1st order nonlinear PDEs for modeling a solid bed cooled by an uprising gas flow, with equations for temperature functions u and v, variable gas velocity function f, and heat transfer coefficient h. The problem also included boundary and initial conditions, and the speaker expressed difficulty in finding a solution. A similar problem with 1st order linear PDEs was also mentioned."
  • #1
raultrigo
2
0
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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  • #2
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,
 

Related to System of 1st order nonlinear PDE's

1. What is a system of 1st order nonlinear PDE's?

A system of 1st order nonlinear PDE's, or partial differential equations, is a set of equations that involve multiple variables and their partial derivatives. These equations are nonlinear, meaning they cannot be represented as a linear combination of the variables and their derivatives.

2. How do you solve a system of 1st order nonlinear PDE's?

Solving a system of 1st order nonlinear PDE's can be a complex process and may require different techniques depending on the specific equations involved. Some common methods include separation of variables, the method of characteristics, or numerical methods such as finite differences or finite elements.

3. What are some real-world applications of systems of 1st order nonlinear PDE's?

Systems of 1st order nonlinear PDE's have many applications in fields such as physics, engineering, and economics. They can be used to model fluid flow, heat transfer, population dynamics, and many other phenomena that involve multiple variables and their rates of change.

4. Can a system of 1st order nonlinear PDE's have more than one solution?

Yes, a system of 1st order nonlinear PDE's can have multiple solutions. In fact, it is common for these systems to have an infinite number of solutions, making it important to carefully consider boundary conditions and initial values in order to find a unique solution.

5. Are there any software tools available for solving systems of 1st order nonlinear PDE's?

Yes, there are several software tools available for solving systems of 1st order nonlinear PDE's, such as MATLAB, Mathematica, and Maple. These programs have built-in functions and solvers specifically designed for solving PDE's and can handle a wide range of equations and boundary conditions.

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