Finding a particular solution to a linear PDE












7












$begingroup$


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







share|cite|improve this question









$endgroup$












  • $begingroup$
    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
















7












$begingroup$


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







share|cite|improve this question









$endgroup$












  • $begingroup$
    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














7












7








7


2



$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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


















  • $begingroup$
    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










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Puebla de Zaragoza

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Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
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Musa

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Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
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