Finding a particular solution to a linear PDE
$begingroup$
I want to solve the PDE
$$frac{partial u}{partial t}+x_1(x_2-x_3) frac{partial u}{partial x_1}+x_2(x_3-x_1) frac{partial u}{partial x_2}+x_3(x_1-x_2) frac{partial u}{partial x_3}=sum_{i=1}^3 alpha_i frac{partial f}{partial x_i}, tag{1} $$
where $alpha_1,alpha_2,alpha_3$ are constants and $f$ is the function
$$f(mathbf{x},t)= frac{alpha _1 left(wp 'left(t;g_2,g_3right)+x_2x_3 left(x_2-x_3right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_2 x_3right)}+frac{alpha_2 left(wp 'left(t;g_2,g_3right)+x_1 x_3 left(x_3-x_1right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_3right)}+frac{alpha _3 left(wp 'left(t;g_2,g_3right)+x_1x_2 left(x_1-x_2right) right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_2right)}+left(alpha _1+alpha _2+alpha _3right) left(zeta left(t;g_2,g_3right)+frac{1}{12} t left(x_1+x_2+x_3right){}^2right). $$
Here $wp$ and $zeta$ are the Weierstraß p- and zeta- functions respectively, with the elliptic invariants
$$ begin{align}
g_2 &= frac{(x_1+x_2+x_3)^4}{12}-2 x_1 x_2 x_3 (x_1+x_2+x_3), \
g_3 &= -(x_1 x_2 x_3)^2+frac{x_1 x_2 x_3 (x_1+x_2+x_3)^3}{6}-frac{(x_1+x_2+x_3)^6}{216}.
end{align} $$
From this point onward the invariants $g_2,g_3$ will not be shown explicitly.
My attempt:
First, I managed to solve the associated homogeneous PDE
$$ frac{partial u_h}{partial t}+x_1(x_2-x_3) frac{partial u_h}{partial x_1}+x_2(x_3-x_1) frac{partial u_h}{partial x_2}+x_3(x_1-x_2) frac{partial u_h}{partial x_3}=0, $$ via the method of characteristics. The solution is given by
$$ u_h =Phi left( X_1(mathbf{x},t), X_2(mathbf{x},t) ,X_3(mathbf{x},t) right) $$
where $Phi$ is an arbitrary function, and
$$X_1=frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{2}x_{3} left(x_{3}-x_{2}right) right)^2}{4 left(wp (t)+ x_{2}x_{3} -frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{2} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_2 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{3} left(x_{1}-x_{3}right) right)^2}{4 left(wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_3 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{2} left(x_{2}-x_{1}right) right)^2}{4 left(wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}. $$
The final ingredient is a particular solution. Denoting the RHS in Equation (1) above by $R(mathbf{x},t)$, Duhamel's principle (or the method of characteristics again) suggests that a particular solution is given by
$$u_p=int_0^t R left( mathbf{X}(mathbf{x},t-u),u right) mathrm{d} u .$$
I tried computing this with Mathematica and it didn't go well. This is probably because Mathematica seems to be unaware of the elliptic identity $wp'^2=4 wp^3-g_2 wp -g_3$.
I would appreciate help with the evaluation of the integral above, or any other method of obtaining a particular solution of Equation (1).
Thank you!
integration ordinary-differential-equations pde closed-form elliptic-functions
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
$endgroup$
add a comment |
$begingroup$
I want to solve the PDE
$$frac{partial u}{partial t}+x_1(x_2-x_3) frac{partial u}{partial x_1}+x_2(x_3-x_1) frac{partial u}{partial x_2}+x_3(x_1-x_2) frac{partial u}{partial x_3}=sum_{i=1}^3 alpha_i frac{partial f}{partial x_i}, tag{1} $$
where $alpha_1,alpha_2,alpha_3$ are constants and $f$ is the function
$$f(mathbf{x},t)= frac{alpha _1 left(wp 'left(t;g_2,g_3right)+x_2x_3 left(x_2-x_3right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_2 x_3right)}+frac{alpha_2 left(wp 'left(t;g_2,g_3right)+x_1 x_3 left(x_3-x_1right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_3right)}+frac{alpha _3 left(wp 'left(t;g_2,g_3right)+x_1x_2 left(x_1-x_2right) right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_2right)}+left(alpha _1+alpha _2+alpha _3right) left(zeta left(t;g_2,g_3right)+frac{1}{12} t left(x_1+x_2+x_3right){}^2right). $$
Here $wp$ and $zeta$ are the Weierstraß p- and zeta- functions respectively, with the elliptic invariants
$$ begin{align}
g_2 &= frac{(x_1+x_2+x_3)^4}{12}-2 x_1 x_2 x_3 (x_1+x_2+x_3), \
g_3 &= -(x_1 x_2 x_3)^2+frac{x_1 x_2 x_3 (x_1+x_2+x_3)^3}{6}-frac{(x_1+x_2+x_3)^6}{216}.
end{align} $$
From this point onward the invariants $g_2,g_3$ will not be shown explicitly.
My attempt:
First, I managed to solve the associated homogeneous PDE
$$ frac{partial u_h}{partial t}+x_1(x_2-x_3) frac{partial u_h}{partial x_1}+x_2(x_3-x_1) frac{partial u_h}{partial x_2}+x_3(x_1-x_2) frac{partial u_h}{partial x_3}=0, $$ via the method of characteristics. The solution is given by
$$ u_h =Phi left( X_1(mathbf{x},t), X_2(mathbf{x},t) ,X_3(mathbf{x},t) right) $$
where $Phi$ is an arbitrary function, and
$$X_1=frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{2}x_{3} left(x_{3}-x_{2}right) right)^2}{4 left(wp (t)+ x_{2}x_{3} -frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{2} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_2 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{3} left(x_{1}-x_{3}right) right)^2}{4 left(wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_3 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{2} left(x_{2}-x_{1}right) right)^2}{4 left(wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}. $$
The final ingredient is a particular solution. Denoting the RHS in Equation (1) above by $R(mathbf{x},t)$, Duhamel's principle (or the method of characteristics again) suggests that a particular solution is given by
$$u_p=int_0^t R left( mathbf{X}(mathbf{x},t-u),u right) mathrm{d} u .$$
I tried computing this with Mathematica and it didn't go well. This is probably because Mathematica seems to be unaware of the elliptic identity $wp'^2=4 wp^3-g_2 wp -g_3$.
I would appreciate help with the evaluation of the integral above, or any other method of obtaining a particular solution of Equation (1).
Thank you!
integration ordinary-differential-equations pde closed-form elliptic-functions
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
$endgroup$
$begingroup$
You already know that a solution exists. This is not enough?
$endgroup$
– timur
Dec 26 '18 at 0:38
$begingroup$
@timur No, I want to see if I can get it explicitly.
$endgroup$
– user1337
Dec 26 '18 at 9:07
add a comment |
$begingroup$
I want to solve the PDE
$$frac{partial u}{partial t}+x_1(x_2-x_3) frac{partial u}{partial x_1}+x_2(x_3-x_1) frac{partial u}{partial x_2}+x_3(x_1-x_2) frac{partial u}{partial x_3}=sum_{i=1}^3 alpha_i frac{partial f}{partial x_i}, tag{1} $$
where $alpha_1,alpha_2,alpha_3$ are constants and $f$ is the function
$$f(mathbf{x},t)= frac{alpha _1 left(wp 'left(t;g_2,g_3right)+x_2x_3 left(x_2-x_3right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_2 x_3right)}+frac{alpha_2 left(wp 'left(t;g_2,g_3right)+x_1 x_3 left(x_3-x_1right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_3right)}+frac{alpha _3 left(wp 'left(t;g_2,g_3right)+x_1x_2 left(x_1-x_2right) right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_2right)}+left(alpha _1+alpha _2+alpha _3right) left(zeta left(t;g_2,g_3right)+frac{1}{12} t left(x_1+x_2+x_3right){}^2right). $$
Here $wp$ and $zeta$ are the Weierstraß p- and zeta- functions respectively, with the elliptic invariants
$$ begin{align}
g_2 &= frac{(x_1+x_2+x_3)^4}{12}-2 x_1 x_2 x_3 (x_1+x_2+x_3), \
g_3 &= -(x_1 x_2 x_3)^2+frac{x_1 x_2 x_3 (x_1+x_2+x_3)^3}{6}-frac{(x_1+x_2+x_3)^6}{216}.
end{align} $$
From this point onward the invariants $g_2,g_3$ will not be shown explicitly.
My attempt:
First, I managed to solve the associated homogeneous PDE
$$ frac{partial u_h}{partial t}+x_1(x_2-x_3) frac{partial u_h}{partial x_1}+x_2(x_3-x_1) frac{partial u_h}{partial x_2}+x_3(x_1-x_2) frac{partial u_h}{partial x_3}=0, $$ via the method of characteristics. The solution is given by
$$ u_h =Phi left( X_1(mathbf{x},t), X_2(mathbf{x},t) ,X_3(mathbf{x},t) right) $$
where $Phi$ is an arbitrary function, and
$$X_1=frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{2}x_{3} left(x_{3}-x_{2}right) right)^2}{4 left(wp (t)+ x_{2}x_{3} -frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{2} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_2 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{3} left(x_{1}-x_{3}right) right)^2}{4 left(wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_3 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{2} left(x_{2}-x_{1}right) right)^2}{4 left(wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}. $$
The final ingredient is a particular solution. Denoting the RHS in Equation (1) above by $R(mathbf{x},t)$, Duhamel's principle (or the method of characteristics again) suggests that a particular solution is given by
$$u_p=int_0^t R left( mathbf{X}(mathbf{x},t-u),u right) mathrm{d} u .$$
I tried computing this with Mathematica and it didn't go well. This is probably because Mathematica seems to be unaware of the elliptic identity $wp'^2=4 wp^3-g_2 wp -g_3$.
I would appreciate help with the evaluation of the integral above, or any other method of obtaining a particular solution of Equation (1).
Thank you!
integration ordinary-differential-equations pde closed-form elliptic-functions
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
$endgroup$
I want to solve the PDE
$$frac{partial u}{partial t}+x_1(x_2-x_3) frac{partial u}{partial x_1}+x_2(x_3-x_1) frac{partial u}{partial x_2}+x_3(x_1-x_2) frac{partial u}{partial x_3}=sum_{i=1}^3 alpha_i frac{partial f}{partial x_i}, tag{1} $$
where $alpha_1,alpha_2,alpha_3$ are constants and $f$ is the function
$$f(mathbf{x},t)= frac{alpha _1 left(wp 'left(t;g_2,g_3right)+x_2x_3 left(x_2-x_3right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_2 x_3right)}+frac{alpha_2 left(wp 'left(t;g_2,g_3right)+x_1 x_3 left(x_3-x_1right)right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_3right)}+frac{alpha _3 left(wp 'left(t;g_2,g_3right)+x_1x_2 left(x_1-x_2right) right)}{2 left(wp left(t;g_2,g_3right)-frac{1}{12} left(x_1+x_2+x_3right){}^2+x_1 x_2right)}+left(alpha _1+alpha _2+alpha _3right) left(zeta left(t;g_2,g_3right)+frac{1}{12} t left(x_1+x_2+x_3right){}^2right). $$
Here $wp$ and $zeta$ are the Weierstraß p- and zeta- functions respectively, with the elliptic invariants
$$ begin{align}
g_2 &= frac{(x_1+x_2+x_3)^4}{12}-2 x_1 x_2 x_3 (x_1+x_2+x_3), \
g_3 &= -(x_1 x_2 x_3)^2+frac{x_1 x_2 x_3 (x_1+x_2+x_3)^3}{6}-frac{(x_1+x_2+x_3)^6}{216}.
end{align} $$
From this point onward the invariants $g_2,g_3$ will not be shown explicitly.
My attempt:
First, I managed to solve the associated homogeneous PDE
$$ frac{partial u_h}{partial t}+x_1(x_2-x_3) frac{partial u_h}{partial x_1}+x_2(x_3-x_1) frac{partial u_h}{partial x_2}+x_3(x_1-x_2) frac{partial u_h}{partial x_3}=0, $$ via the method of characteristics. The solution is given by
$$ u_h =Phi left( X_1(mathbf{x},t), X_2(mathbf{x},t) ,X_3(mathbf{x},t) right) $$
where $Phi$ is an arbitrary function, and
$$X_1=frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{2}x_{3} left(x_{3}-x_{2}right) right)^2}{4 left(wp (t)+ x_{2}x_{3} -frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{2} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_2 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{3} left(x_{1}-x_{3}right) right)^2}{4 left(wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+x_{1} x_{3}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}, \
X_3 = frac{12 x_1 x_2 x_3}{left(x_1+x_2+x_3right)^2-12 left(frac{left(wp'(t)-x_{1} x_{2} left(x_{2}-x_{1}right) right)^2}{4 left(wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)^2}-wp (t)+ x_{1} x_{2}-frac{1}{12} left(x_1+x_2+x_3right)^2right)}. $$
The final ingredient is a particular solution. Denoting the RHS in Equation (1) above by $R(mathbf{x},t)$, Duhamel's principle (or the method of characteristics again) suggests that a particular solution is given by
$$u_p=int_0^t R left( mathbf{X}(mathbf{x},t-u),u right) mathrm{d} u .$$
I tried computing this with Mathematica and it didn't go well. This is probably because Mathematica seems to be unaware of the elliptic identity $wp'^2=4 wp^3-g_2 wp -g_3$.
I would appreciate help with the evaluation of the integral above, or any other method of obtaining a particular solution of Equation (1).
Thank you!
integration ordinary-differential-equations pde closed-form elliptic-functions
integration ordinary-differential-equations pde closed-form elliptic-functions
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
asked Dec 17 '18 at 16:21
user1337user1337
16.8k43592
16.8k43592
$begingroup$
You already know that a solution exists. This is not enough?
$endgroup$
– timur
Dec 26 '18 at 0:38
$begingroup$
@timur No, I want to see if I can get it explicitly.
$endgroup$
– user1337
Dec 26 '18 at 9:07
add a comment |
$begingroup$
You already know that a solution exists. This is not enough?
$endgroup$
– timur
Dec 26 '18 at 0:38
$begingroup$
@timur No, I want to see if I can get it explicitly.
$endgroup$
– user1337
Dec 26 '18 at 9:07
$begingroup$
You already know that a solution exists. This is not enough?
$endgroup$
– timur
Dec 26 '18 at 0:38
$begingroup$
You already know that a solution exists. This is not enough?
$endgroup$
– timur
Dec 26 '18 at 0:38
$begingroup$
@timur No, I want to see if I can get it explicitly.
$endgroup$
– user1337
Dec 26 '18 at 9:07
$begingroup$
@timur No, I want to see if I can get it explicitly.
$endgroup$
– user1337
Dec 26 '18 at 9:07
add a comment |
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$begingroup$
You already know that a solution exists. This is not enough?
$endgroup$
– timur
Dec 26 '18 at 0:38
$begingroup$
@timur No, I want to see if I can get it explicitly.
$endgroup$
– user1337
Dec 26 '18 at 9:07