How to Integrate and Compare Solutions for a Partial Differential System?

In summary, a partial differential system is a mathematical model used to describe systems that change over both space and time. It involves partial derivatives and has applications in various fields such as physics, engineering, and economics. The main difference between partial differential systems and ordinary differential equations is the number of independent variables, and they can be solved using various methods such as separation of variables and finite difference methods. Working with partial differential systems can be challenging due to their complexity and the required understanding of mathematical concepts and underlying principles.
  • #1
arrow27
7
0
\begin{array}{l}
u = u(x,y) \\
v = v(x,y) \\
and\\
{u_x} + 4{v_y} = 0 \\
{v_x} + 9{u_y} = 0 \\
with\ the\ initial\ conditions \\
u(x,0) = 2x _(3)\\
v(x,0) = 3x _(4)\\
\end{array}

Easy,
[tex]u_{xx}-36u_{yy}=0[/tex] and [tex]v_{xx}-36v_{yy}=0[/tex]
General solution [tex]u\left ( x,y \right )=h\left ( x+6y \right )+g\left ( y-6x \right )[/tex]
Similar,
[tex]v\left ( x,y \right )=h\left ( x+6y \right )+g\left ( y-6x \right )[/tex]

From (3) : [tex]2x=h\left ( 6x \right )+g\left ( -6x \right )[/tex]
From (4) : [tex]3x=h\left ( 6x \right )+g\left ( -6x \right )[/tex]

How to continue?
 
Last edited by a moderator:
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  • #2
arrow27 said:
\begin{array}{l}
u = u(x,y) \\
v = v(x,y) \\
and\\
{u_x} + 4{v_y} = 0 \\
{v_x} + 9{u_y} = 0 \\
with\ the\ initial\ conditions \\
u(x,0) = 2x _(3)\\
v(x,0) = 3x _(4)\\
\end{array}

Easy,
[tex]u_{xx}-36u_{yy}=0[/tex] and [tex]v_{xx}-36v_{yy}=0[/tex]
General solution [tex]u\left ( x,y \right )=h\left ( x+6y \right )+g\left ( y-6x \right )[/tex]
Similar,
[tex]v\left ( x,y \right )=h\left ( x+6y \right )+g\left ( y-6x \right )[/tex]

From (3) : [tex]2x=h\left ( 6x \right )+g\left ( -6x \right )[/tex]
From (4) : [tex]3x=h\left ( 6x \right )+g\left ( -6x \right )[/tex]

How to continue?
Why are you assuming the solutions $u$ and $v$ are exactly the same? As you're seeing this is inconsistent with the initial conditions.
 
  • #3
What can i do?
 
  • #4
arrow27 said:
What can i do?
Choose one of them, say \(\displaystyle u = g(x+6y) + g(y-6x)\), then sub this back into your system. Then integrate each giving the solution for \(\displaystyle v(x,y)\). Then use your BC's.
 
  • #5
But we have u_x and u_x in equations.
For example,

[tex]\[u = {g'(x)}(x + 6y) - 6{g'(x)}(x - 6y)\] [/tex]
Sub this in the first equation?
 
Last edited:
  • #6
arrow27 said:
But we have u_x and u_x in equations.
For example,

[tex]\[u = {g'(x)}(x + 6y) - 6{g'(x)}(x - 6y)\] [/tex]
Sub this in the first equation?
Sorry, that was a typo. If

\(\displaystyle u = h(6x+y) + g(6x-y)\)

then

\(\displaystyle u_x = 6h'(6x+y) + 6g'(6x-y)\)

\(\displaystyle u_y = h'(6x+y) - g'(6x-y)\)

So from the original set of PDEs we have

\(\displaystyle 6h'(6x+y) + 6g'(6x - y) + 4 v_y = 0\)

\(\displaystyle v_x + 9\left(h'(6x+y) - g'(6x-y)\right) = 0\)

or

\(\displaystyle v_x = - 9h'(6x+y) + 9 g'(6x-y)\)

\(\displaystyle v_y = - \dfrac{3}{2} h'(6x+y) - \dfrac{3}{2} g'(6x - y). \)

Now integrate each separately and compare.
 

Related to How to Integrate and Compare Solutions for a Partial Differential System?

1. What is a partial differential system?

A partial differential system is a mathematical model used to describe the behavior of systems that change over both space and time. It consists of a set of equations that involve partial derivatives, which represent the rate of change of a variable with respect to multiple independent variables.

2. What are the applications of partial differential systems?

Partial differential systems have a wide range of applications in various fields such as physics, engineering, and economics. They are commonly used to model physical phenomena such as fluid flow, heat transfer, and electromagnetic fields, as well as to solve optimization problems and make predictions in financial markets.

3. What is the difference between partial differential systems and ordinary differential equations?

The main difference between partial differential systems and ordinary differential equations is that the former involve multiple independent variables, while the latter only involve a single independent variable. This makes partial differential systems more complex and often requires advanced techniques for solving them.

4. How are partial differential systems solved?

Partial differential systems can be solved using various numerical and analytical methods. Some commonly used techniques include separation of variables, finite difference methods, and numerical approximation methods such as finite element analysis and finite volume methods.

5. What are the challenges of working with partial differential systems?

One of the main challenges of working with partial differential systems is their complexity, as they involve multiple variables and can be difficult to solve analytically. They also require a strong understanding of mathematical concepts and techniques, as well as a good understanding of the underlying physical or engineering principles being modeled.

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