Why is the characteristic of (d/dx) + (d/dt) = 0 not c = x + t?

Thread starter Snippy
Start date Jun 12, 2010
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2nd order Pde

In summary, a 2nd order PDE, or second order partial differential equation, is a mathematical equation involving partial derivatives of a function with respect to two or more independent variables. The characteristics of a 2nd order PDE include its order, type, linearity, and boundary conditions. There are various methods for solving 2nd order PDEs, and they have many applications in physics and engineering. 2nd order PDEs differ from 1st order PDEs in that they involve second order derivatives and have more restrictive boundary conditions.

Jun 12, 2010

Snippy

Why is the characteristic of

(d/dx) + (d/dt) = 0 where d is small delta

c = x - t and not c = x + t

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Jun 13, 2010

elibj123

Because[tex](\partial_{x}+\partial_{t})f(x+t)=2f'(x+t)[/tex]

While

[tex](\partial_{x}+\partial_{t})f(x-t)=f'(x-t)-f'(x-t)=0[/tex]

So you need solutions of the latter form u(x,t)=f(x-t), which means x-t=const are charcteristic lines.

Related to Why is the characteristic of (d/dx) + (d/dt) = 0 not c = x + t?

1. What is a 2nd order PDE?

A 2nd order PDE, or second order partial differential equation, is a mathematical equation that involves the partial derivatives of a function with respect to two or more independent variables. It is a type of differential equation commonly used in physics and engineering to model various phenomena.

2. What are the characteristics of a 2nd order PDE?

The characteristics of a 2nd order PDE include the order, type, linearity, and boundary conditions. The order refers to the highest derivative present in the equation, the type refers to whether it is elliptic, parabolic, or hyperbolic, and linearity refers to whether the equation is linear or nonlinear. The boundary conditions specify the values of the function at the boundaries of the domain.

3. How are 2nd order PDEs solved?

There are various methods for solving 2nd order PDEs, including separation of variables, the method of characteristics, and numerical methods such as finite difference or finite element methods. The appropriate method depends on the type of PDE and the boundary conditions.

4. What are some applications of 2nd order PDEs?

2nd order PDEs have many applications in physics and engineering, including heat transfer, fluid dynamics, electromagnetism, and quantum mechanics. They can be used to model and predict the behavior of various physical systems and phenomena.

5. How do 2nd order PDEs differ from 1st order PDEs?

The main difference between 2nd order and 1st order PDEs is that 2nd order PDEs involve second order partial derivatives, while 1st order PDEs only involve first order partial derivatives. This means that 2nd order PDEs are more complex and can model more intricate systems. Additionally, the boundary conditions for 2nd order PDEs are more restrictive, making them more challenging to solve.