hi all...

i have 2 Q:

1-what is the diffrent between Quasi-Linear pde and Linear pde ?

2- How we can solve Quasi-Linear ??

any body can help !!!:confused:

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- Aug 18th 2007, 07:22 PMsweetQuasi-Linear
hi all...

i have 2 Q:

1-what is the diffrent between Quasi-Linear pde and Linear pde ?

2- How we can solve Quasi-Linear ??

any body can help !!!:confused: - Aug 18th 2007, 07:31 PMThePerfectHacker
Let solve a PDE, if it happens that , i.e. any linear combination is a solution, then we say the PDE is

*linear*.

For example, consider the wave equation,

.

Confirm that if are solutions then is a solution as well.

Another way to think of linear is that , the solution, is just like what linear means in an ordinary differencial equation. So is linear. But is not, because by analogy we never have being considered linear ODE.

Quasi-linear means (this is how I understand it) that the**highest order terms**are linear.

For example,

Is not linear but it is quasi-linear. Because the highest order terms are linear. - Aug 18th 2007, 07:35 PMThePerfectHackerQuote:

2- How we can solve Quasi-Linear ??

Depending on the sign of : positive, negative, zero. We have 3 cases: hyperbolic, elliptic, parabolic. Each one can be simplified by a method knows as**charachteristics curves**. When properly reduced by appropriate substitutions we get the "canonical form". From there we use certain PDE methods to solve that. - Aug 18th 2007, 07:51 PMsweetQuote:

Quasi-linear means (this is how I understand it) that the highest order terms are linear.

For example,

u_{xx}+\cos(x+y)u_{yy}+u_{xy}=e^{x+y} u_x\cdot u_y^2

Is not linear but it is quasi-linear. Because the highest order terms u_{xx},u_{yy},u_{xy} are linear.

but how can i solv quasi-linear of the first order..? - Aug 18th 2007, 07:58 PMThePerfectHacker
Here is Lagrange's method from 1772 (older than America):eek:

Consider the equation,

.

Note, this is not a linear equation because contain , the solution, in them but still we can solve it.

We do this by solving, (I do not like using differencials but here they play a nice role),

.

Let solve the differencial above (not the actual equation). Then the solution to the actual PDE is given by . - Aug 18th 2007, 08:11 PMsweet
thank u :):)

do u have a paper of the Lagrange's method - Aug 18th 2007, 08:37 PMThePerfectHacker