Skip to main content
MathOverflow

Return to Question

added 237 characters in body; edited title
Source Link
FraSchelle
FraSchelle
  • 245
  • 1
  • 7

Analytic solution of a system of semi-linearlinear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form

$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+\mathbf{A}\partial_{t}\mathbf{u}+\mathbf{B}\left(t\right)\mathbf{u}=0$$

with $\left(x,y,t\right)$-independent matrix $\mathbf{A}$ and $t$-dependent / $\left(x,y\right)$-independent $\mathbf{B}$. Also, $t\in\left[0,2\pi\right]$ is not really a time, but an angle, and $\mathbf{B}\left(t+2\pi\right)=\mathbf{B}\left(t\right)$ is periodic, so I guess the solutions will be periodic as well. Actually $\mathbf{A}$ is real and diagonal, so my system of equations seems to be in a canonical form from the beginning. Note nevertheless that two eigenvalues of $$\mathbf{A}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$$ are degenerate, so I wonder if the system is still hyperbolic (I know it is no more strictly hyperbolic according to Courant (see references below), but Wikipedia defines it as hyperbolic / I've been unable to find the Wikipedia definition in any textbook I've put an hand on). I would like (if possible) to obtain analytic (in a close form or as a series expansion) solutions of this problem, or understand a bit better which perturbation scheme I could use. It seems to me this system is extraordinary simple, and at the same time I'm paralysed by my ignorance in powerful enough methods.

So my first and general question would be : which literature would you recommend to me ? about this problem.

Giving more details: for the moment I've found

R. Courant and P. Lax On nonlinear partial differential equations with two independent variables Commun. Pure Appl. Math. 2, 255 (1949). (beyond a paywall)

where section 3 seems of interest for me, it shortly discusses an old method by

O. Perron Über Existenz und Nichtexistenz von Integralen partieller Differentialgleichungssysteme im reellen Gebiet Math. Zeitschrift 27, 549 (1928). (also beyond a paywall)

which seems to answer my problem, giving analytical solution of system of semi-linear partial differential equation. The book by

R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II: Partial Differential Equation (John Wiley and Sons, 1962).

seems too much related to non-linear systems, and I've been unable to figure out what to do with my problem. Finally a short lecture note by

E. Kersalé, Analytic Solutions of Partial Differential Equations http://www1.maths.leeds.ac.uk/~kersale/teaching.html (2003)

ends up with a short discussion of my problem, but I still do not figure out what to do with the system of ordinary differential equations once fond the characteristics lines / surfaces... (actually, two characteristics are circles, and two straight linelines are degenerate). So are there other (perhaps better or more specific for physicists) textbook/notes you would recommend ?

Thanks in advance for any remark aiming at improving this question.

Analytic solution of a system of semi-linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form

$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+\mathbf{A}\partial_{t}\mathbf{u}+\mathbf{B}\left(t\right)\mathbf{u}=0$$

with $\left(x,y,t\right)$-independent matrix $\mathbf{A}$ and $t$-dependent / $\left(x,y\right)$-independent $\mathbf{B}$. Actually $\mathbf{A}$ is real and diagonal, so my system of equations seems to be in a canonical form from the beginning. Note nevertheless that two eigenvalues of $$\mathbf{A}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$$ are degenerate, so I wonder if the system is still hyperbolic (I know it is no more strictly hyperbolic according to Courant (see references below), but Wikipedia defines it as hyperbolic / I've been unable to find the Wikipedia definition in any textbook I've put an hand on). I would like (if possible) to obtain analytic solutions of this problem, or understand a bit better which perturbation scheme I could use. It seems to me this system is extraordinary simple, and at the same time I'm paralysed by my ignorance in powerful enough methods.

So my first and general question would be : which literature would you recommend to me ? about this problem.

Giving more details: for the moment I've found

R. Courant and P. Lax On nonlinear partial differential equations with two independent variables Commun. Pure Appl. Math. 2, 255 (1949). (beyond a paywall)

where section 3 seems of interest for me, it shortly discusses an old method by

O. Perron Über Existenz und Nichtexistenz von Integralen partieller Differentialgleichungssysteme im reellen Gebiet Math. Zeitschrift 27, 549 (1928). (also beyond a paywall)

which seems to answer my problem, giving analytical solution of system of semi-linear partial differential equation. The book by

R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II: Partial Differential Equation (John Wiley and Sons, 1962).

seems too much related to non-linear systems, and I've been unable to figure out what to do with my problem. Finally a short lecture note by

E. Kersalé, Analytic Solutions of Partial Differential Equations http://www1.maths.leeds.ac.uk/~kersale/teaching.html (2003)

ends up with a short discussion of my problem, but I still do not figure out what to do with the system of ordinary differential equations once fond the characteristics lines / surfaces... (actually, two characteristics are circles, and two straight line are degenerate). So are there other (perhaps better or more specific for physicists) textbook/notes you would recommend ?

Thanks in advance for any remark aiming at improving this question.

Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form

$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+\mathbf{A}\partial_{t}\mathbf{u}+\mathbf{B}\left(t\right)\mathbf{u}=0$$

with $\left(x,y,t\right)$-independent matrix $\mathbf{A}$ and $t$-dependent / $\left(x,y\right)$-independent $\mathbf{B}$. Also, $t\in\left[0,2\pi\right]$ is not really a time, but an angle, and $\mathbf{B}\left(t+2\pi\right)=\mathbf{B}\left(t\right)$ is periodic, so I guess the solutions will be periodic as well. Actually $\mathbf{A}$ is real and diagonal, so my system of equations seems to be in a canonical form from the beginning. Note nevertheless that two eigenvalues of $$\mathbf{A}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$$ are degenerate, so I wonder if the system is still hyperbolic (I know it is no more strictly hyperbolic according to Courant (see references below), but Wikipedia defines it as hyperbolic / I've been unable to find the Wikipedia definition in any textbook I've put an hand on). I would like (if possible) to obtain analytic (in a close form or as a series expansion) solutions of this problem, or understand a bit better which perturbation scheme I could use. It seems to me this system is extraordinary simple, and at the same time I'm paralysed by my ignorance in powerful enough methods.

So my first and general question would be : which literature would you recommend to me ? about this problem.

Giving more details: for the moment I've found

R. Courant and P. Lax On nonlinear partial differential equations with two independent variables Commun. Pure Appl. Math. 2, 255 (1949). (beyond a paywall)

where section 3 seems of interest for me, it shortly discusses an old method by

O. Perron Über Existenz und Nichtexistenz von Integralen partieller Differentialgleichungssysteme im reellen Gebiet Math. Zeitschrift 27, 549 (1928). (also beyond a paywall)

which seems to answer my problem, giving analytical solution of system of semi-linear partial differential equation. The book by

R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II: Partial Differential Equation (John Wiley and Sons, 1962).

seems too much related to non-linear systems, and I've been unable to figure out what to do with my problem. Finally a short lecture note by

E. Kersalé, Analytic Solutions of Partial Differential Equations http://www1.maths.leeds.ac.uk/~kersale/teaching.html (2003)

ends up with a short discussion of my problem, but I still do not figure out what to do with the system of ordinary differential equations once fond the characteristics lines / surfaces... (actually, two characteristics are circles, and two straight lines are degenerate). So are there other (perhaps better or more specific for physicists) textbook/notes you would recommend ?

Thanks in advance for any remark aiming at improving this question.

Source Link
FraSchelle
FraSchelle
  • 245
  • 1
  • 7

Analytic solution of a system of semi-linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form

$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+\mathbf{A}\partial_{t}\mathbf{u}+\mathbf{B}\left(t\right)\mathbf{u}=0$$

with $\left(x,y,t\right)$-independent matrix $\mathbf{A}$ and $t$-dependent / $\left(x,y\right)$-independent $\mathbf{B}$. Actually $\mathbf{A}$ is real and diagonal, so my system of equations seems to be in a canonical form from the beginning. Note nevertheless that two eigenvalues of $$\mathbf{A}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$$ are degenerate, so I wonder if the system is still hyperbolic (I know it is no more strictly hyperbolic according to Courant (see references below), but Wikipedia defines it as hyperbolic / I've been unable to find the Wikipedia definition in any textbook I've put an hand on). I would like (if possible) to obtain analytic solutions of this problem, or understand a bit better which perturbation scheme I could use. It seems to me this system is extraordinary simple, and at the same time I'm paralysed by my ignorance in powerful enough methods.

So my first and general question would be : which literature would you recommend to me ? about this problem.

Giving more details: for the moment I've found

R. Courant and P. Lax On nonlinear partial differential equations with two independent variables Commun. Pure Appl. Math. 2, 255 (1949). (beyond a paywall)

where section 3 seems of interest for me, it shortly discusses an old method by

O. Perron Über Existenz und Nichtexistenz von Integralen partieller Differentialgleichungssysteme im reellen Gebiet Math. Zeitschrift 27, 549 (1928). (also beyond a paywall)

which seems to answer my problem, giving analytical solution of system of semi-linear partial differential equation. The book by

R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II: Partial Differential Equation (John Wiley and Sons, 1962).

seems too much related to non-linear systems, and I've been unable to figure out what to do with my problem. Finally a short lecture note by

E. Kersalé, Analytic Solutions of Partial Differential Equations http://www1.maths.leeds.ac.uk/~kersale/teaching.html (2003)

ends up with a short discussion of my problem, but I still do not figure out what to do with the system of ordinary differential equations once fond the characteristics lines / surfaces... (actually, two characteristics are circles, and two straight line are degenerate). So are there other (perhaps better or more specific for physicists) textbook/notes you would recommend ?

Thanks in advance for any remark aiming at improving this question.