Linear second order partial differential equation with variable coefficients
up vote
0
down vote
favorite
So after using the method of separation of variables of a PDE in 3 variables, I have simplified it to a ODE in time and a PDE of two variables.
However, my resulting PDE is a linear second order partial differential equation with variable coefficients of the form
$Au_{xx} + Bu_{yy} + C(x)u_x + D(y)u_y + (p(x,y) + E)u = 0$
where A,B, and E are constants, p(x,y) is a polynomial of x and y of degree 2, and C(x) and D(y) are functions of x and y respectively. Any suggestions on how to solve the resulting PDE?
analysis pde linear-pde
asked Nov 15 at 17:46
Ken Klark
91
add a comment |
up vote
0
down vote
favorite
So after using the method of separation of variables of a PDE in 3 variables, I have simplified it to a ODE in time and a PDE of two variables.
However, my resulting PDE is a linear second order partial differential equation with variable coefficients of the form
$Au_{xx} + Bu_{yy} + C(x)u_x + D(y)u_y + (p(x,y) + E)u = 0$
where A,B, and E are constants, p(x,y) is a polynomial of x and y of degree 2, and C(x) and D(y) are functions of x and y respectively. Any suggestions on how to solve the resulting PDE?
analysis pde linear-pde
asked Nov 15 at 17:46
Ken Klark
91
Do you have any reason to think that the values of $A$ and $B$ are such that your PDE is elliptic ? This is probably an easier framework to work with.
– Gâteau-Gallois
Nov 16 at 8:29
Yes, definitely. I forgot to mention it, but the PDE is elliptic on a bounded rectangle (of some sort). I was wondering if I am able to solve is using the separation of variables again with u = v(x)w(y)? If doing so I end up with a ODE for x on one side and an ODE for y on the other, but they are both depending on the common polynomial p(x,y). @Gâteau-Gallois
– Ken Klark
Nov 16 at 14:53
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So after using the method of separation of variables of a PDE in 3 variables, I have simplified it to a ODE in time and a PDE of two variables.
However, my resulting PDE is a linear second order partial differential equation with variable coefficients of the form
$Au_{xx} + Bu_{yy} + C(x)u_x + D(y)u_y + (p(x,y) + E)u = 0$
where A,B, and E are constants, p(x,y) is a polynomial of x and y of degree 2, and C(x) and D(y) are functions of x and y respectively. Any suggestions on how to solve the resulting PDE?
analysis pde linear-pde
asked Nov 15 at 17:46
Ken Klark
91
So after using the method of separation of variables of a PDE in 3 variables, I have simplified it to a ODE in time and a PDE of two variables.
However, my resulting PDE is a linear second order partial differential equation with variable coefficients of the form
$Au_{xx} + Bu_{yy} + C(x)u_x + D(y)u_y + (p(x,y) + E)u = 0$
where A,B, and E are constants, p(x,y) is a polynomial of x and y of degree 2, and C(x) and D(y) are functions of x and y respectively. Any suggestions on how to solve the resulting PDE?
analysis pde linear-pde
analysis pde linear-pde
asked Nov 15 at 17:46
Ken Klark
91
asked Nov 15 at 17:46
Ken Klark
91
asked Nov 15 at 17:46
Ken Klark
91
asked Nov 15 at 17:46
Ken Klark
91
asked Nov 15 at 17:46
Ken Klark
91
91
Do you have any reason to think that the values of $A$ and $B$ are such that your PDE is elliptic ? This is probably an easier framework to work with.
– Gâteau-Gallois
Nov 16 at 8:29
Yes, definitely. I forgot to mention it, but the PDE is elliptic on a bounded rectangle (of some sort). I was wondering if I am able to solve is using the separation of variables again with u = v(x)w(y)? If doing so I end up with a ODE for x on one side and an ODE for y on the other, but they are both depending on the common polynomial p(x,y). @Gâteau-Gallois
– Ken Klark
Nov 16 at 14:53
add a comment |
Do you have any reason to think that the values of $A$ and $B$ are such that your PDE is elliptic ? This is probably an easier framework to work with.
– Gâteau-Gallois
Nov 16 at 8:29
Yes, definitely. I forgot to mention it, but the PDE is elliptic on a bounded rectangle (of some sort). I was wondering if I am able to solve is using the separation of variables again with u = v(x)w(y)? If doing so I end up with a ODE for x on one side and an ODE for y on the other, but they are both depending on the common polynomial p(x,y). @Gâteau-Gallois
– Ken Klark
Nov 16 at 14:53
Do you have any reason to think that the values of $A$ and $B$ are such that your PDE is elliptic ? This is probably an easier framework to work with.
– Gâteau-Gallois
Nov 16 at 8:29
Do you have any reason to think that the values of $A$ and $B$ are such that your PDE is elliptic ? This is probably an easier framework to work with.
– Gâteau-Gallois
Nov 16 at 8:29
Yes, definitely. I forgot to mention it, but the PDE is elliptic on a bounded rectangle (of some sort). I was wondering if I am able to solve is using the separation of variables again with u = v(x)w(y)? If doing so I end up with a ODE for x on one side and an ODE for y on the other, but they are both depending on the common polynomial p(x,y). @Gâteau-Gallois
– Ken Klark
Nov 16 at 14:53
Yes, definitely. I forgot to mention it, but the PDE is elliptic on a bounded rectangle (of some sort). I was wondering if I am able to solve is using the separation of variables again with u = v(x)w(y)? If doing so I end up with a ODE for x on one side and an ODE for y on the other, but they are both depending on the common polynomial p(x,y). @Gâteau-Gallois
– Ken Klark
Nov 16 at 14:53
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3000018%2flinear-second-order-partial-differential-equation-with-variable-coefficients%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Do you have any reason to think that the values of $A$ and $B$ are such that your PDE is elliptic ? This is probably an easier framework to work with.
– Gâteau-Gallois
Nov 16 at 8:29
Yes, definitely. I forgot to mention it, but the PDE is elliptic on a bounded rectangle (of some sort). I was wondering if I am able to solve is using the separation of variables again with u = v(x)w(y)? If doing so I end up with a ODE for x on one side and an ODE for y on the other, but they are both depending on the common polynomial p(x,y). @Gâteau-Gallois
– Ken Klark
Nov 16 at 14:53