Could this linear homogeneous 1D transport equation with variable coefficient yield a solution that can be...












0














Note: this question was edited to reflect an evolving insight on part of the author. However, the original question still stands (although it has hopefully been formulated more clearly now).





For a physics problem that interests me, I am looking for a solution to the following linear homogeneous 1D transport equation with variable coefficient:



$$
f^{(1,0)}(t,x) - (g_1(t),h_1(x) + g_2(t),h_2(x));f^{(0,1)}(t,x) = 0
$$



In this PDE, all functions and variables are real-valued. $x$ has the meaning of a spatial coordinate and $t$ has the meaning of time. Superscripts between parentheses denote order of differentiation w.r.t. respective arguments. While the two functions of time $g_1(t)$ and $g_2(t)$ can best be considered unspecified1, both functions of space $h_1(x)$ and $h_2(x)$ are given by simple algebraic expressions:




  • $h_1(x)=+x(x^gamma-1)$


  • $h_2(x)=-x(x-1)$,


where $gamma$ is a positive real number. Since this is a transport equation, I have learned that the method of characteristics can be applied to transform a PDE into an ODE. As far as I have understood, I can write for the characteristic $X(t)$:



$$
frac{d}{dt};f(X(t),t) = 0
$$



$$
frac{dX}{dt} = - g_1(t),h_1(X(t)) - g_2(t),h_2(X(t))
$$



$$
f(t,x) = F(X^*(t,x))
$$



here, $X^*(t,x)$ is a solution to the characteristic equation above subject to initial conditions $X(0) = X^*(0,x) = x$. $F(x)$ is an unknown function.





Of course a transport equation will transport any signal we give it. Clearly though, an initial signal equal to e.g. sine of $x$ will not remain a sine of $x$ for long. My question is therefore: given $mathbf{h_1(x)}$ and $mathbf{h_2(x)}$ (but not given $mathbf{g_1(t)}$ and $mathbf{g_2(t)}$), can we say anything about the functional form that $mathbf{F(x)}$ must have for it to be transportable analytically?



There are a few constraints that $f(t,x)$ should adhere to:




  • The domain of $f(t,x)$ is $[0,infty)$, $(0,1]$ and its codomain is $[0,infty)$.

  • $f(t,1)=0$


  • $lim_{x downarrow 0} f(t,x) = +infty$.


  • $f(t,x)$ is a monotonically decreasing function of $x$.

  • $lim_{x downarrow 0} f^{(0,1)}(t,x) = -infty$

  • $lim_{x uparrow 1} f^{(0,1)}(t,x) = -infty$


In short, the inverse function of $f(t,x)$ w.r.t. its second argument could look like a Gaussian or a Lorentzian. In fact, the PDE considered here was derived from a much more horrible looking PDE formulated in terms of this inverse, as well as some other considerations.





1 They each depend on their own set of nonlinear evolution equations which I'm quite sure do not have a closed-form solution or approximation. To pull these equations into the context of this question would in all likelihood not improve it, I feel.










share|cite|improve this question
























  • What's non-linear about this equation? You have $frac{partial f}{partial t} = a(x,t)frac{partial f}{partial x}$
    – Dylan
    Nov 30 '18 at 12:17












  • You are correct. I have changed the title and text accordingly.
    – Casper Pranger
    Nov 30 '18 at 13:28










  • Thanks to the remark of @Dylan that this is not a non-linear equation at all, I now realize that its classification would be 'one-dimensional transport equation with variable coefficient'. I am currently looking into applying the method of characteristics, though so far I can not say that it is very helpful to me. If I find anything concrete, I will be sure to update the question.
    – Casper Pranger
    Nov 30 '18 at 17:40
















0














Note: this question was edited to reflect an evolving insight on part of the author. However, the original question still stands (although it has hopefully been formulated more clearly now).





For a physics problem that interests me, I am looking for a solution to the following linear homogeneous 1D transport equation with variable coefficient:



$$
f^{(1,0)}(t,x) - (g_1(t),h_1(x) + g_2(t),h_2(x));f^{(0,1)}(t,x) = 0
$$



In this PDE, all functions and variables are real-valued. $x$ has the meaning of a spatial coordinate and $t$ has the meaning of time. Superscripts between parentheses denote order of differentiation w.r.t. respective arguments. While the two functions of time $g_1(t)$ and $g_2(t)$ can best be considered unspecified1, both functions of space $h_1(x)$ and $h_2(x)$ are given by simple algebraic expressions:




  • $h_1(x)=+x(x^gamma-1)$


  • $h_2(x)=-x(x-1)$,


where $gamma$ is a positive real number. Since this is a transport equation, I have learned that the method of characteristics can be applied to transform a PDE into an ODE. As far as I have understood, I can write for the characteristic $X(t)$:



$$
frac{d}{dt};f(X(t),t) = 0
$$



$$
frac{dX}{dt} = - g_1(t),h_1(X(t)) - g_2(t),h_2(X(t))
$$



$$
f(t,x) = F(X^*(t,x))
$$



here, $X^*(t,x)$ is a solution to the characteristic equation above subject to initial conditions $X(0) = X^*(0,x) = x$. $F(x)$ is an unknown function.





Of course a transport equation will transport any signal we give it. Clearly though, an initial signal equal to e.g. sine of $x$ will not remain a sine of $x$ for long. My question is therefore: given $mathbf{h_1(x)}$ and $mathbf{h_2(x)}$ (but not given $mathbf{g_1(t)}$ and $mathbf{g_2(t)}$), can we say anything about the functional form that $mathbf{F(x)}$ must have for it to be transportable analytically?



There are a few constraints that $f(t,x)$ should adhere to:




  • The domain of $f(t,x)$ is $[0,infty)$, $(0,1]$ and its codomain is $[0,infty)$.

  • $f(t,1)=0$


  • $lim_{x downarrow 0} f(t,x) = +infty$.


  • $f(t,x)$ is a monotonically decreasing function of $x$.

  • $lim_{x downarrow 0} f^{(0,1)}(t,x) = -infty$

  • $lim_{x uparrow 1} f^{(0,1)}(t,x) = -infty$


In short, the inverse function of $f(t,x)$ w.r.t. its second argument could look like a Gaussian or a Lorentzian. In fact, the PDE considered here was derived from a much more horrible looking PDE formulated in terms of this inverse, as well as some other considerations.





1 They each depend on their own set of nonlinear evolution equations which I'm quite sure do not have a closed-form solution or approximation. To pull these equations into the context of this question would in all likelihood not improve it, I feel.










share|cite|improve this question
























  • What's non-linear about this equation? You have $frac{partial f}{partial t} = a(x,t)frac{partial f}{partial x}$
    – Dylan
    Nov 30 '18 at 12:17












  • You are correct. I have changed the title and text accordingly.
    – Casper Pranger
    Nov 30 '18 at 13:28










  • Thanks to the remark of @Dylan that this is not a non-linear equation at all, I now realize that its classification would be 'one-dimensional transport equation with variable coefficient'. I am currently looking into applying the method of characteristics, though so far I can not say that it is very helpful to me. If I find anything concrete, I will be sure to update the question.
    – Casper Pranger
    Nov 30 '18 at 17:40














0












0








0







Note: this question was edited to reflect an evolving insight on part of the author. However, the original question still stands (although it has hopefully been formulated more clearly now).





For a physics problem that interests me, I am looking for a solution to the following linear homogeneous 1D transport equation with variable coefficient:



$$
f^{(1,0)}(t,x) - (g_1(t),h_1(x) + g_2(t),h_2(x));f^{(0,1)}(t,x) = 0
$$



In this PDE, all functions and variables are real-valued. $x$ has the meaning of a spatial coordinate and $t$ has the meaning of time. Superscripts between parentheses denote order of differentiation w.r.t. respective arguments. While the two functions of time $g_1(t)$ and $g_2(t)$ can best be considered unspecified1, both functions of space $h_1(x)$ and $h_2(x)$ are given by simple algebraic expressions:




  • $h_1(x)=+x(x^gamma-1)$


  • $h_2(x)=-x(x-1)$,


where $gamma$ is a positive real number. Since this is a transport equation, I have learned that the method of characteristics can be applied to transform a PDE into an ODE. As far as I have understood, I can write for the characteristic $X(t)$:



$$
frac{d}{dt};f(X(t),t) = 0
$$



$$
frac{dX}{dt} = - g_1(t),h_1(X(t)) - g_2(t),h_2(X(t))
$$



$$
f(t,x) = F(X^*(t,x))
$$



here, $X^*(t,x)$ is a solution to the characteristic equation above subject to initial conditions $X(0) = X^*(0,x) = x$. $F(x)$ is an unknown function.





Of course a transport equation will transport any signal we give it. Clearly though, an initial signal equal to e.g. sine of $x$ will not remain a sine of $x$ for long. My question is therefore: given $mathbf{h_1(x)}$ and $mathbf{h_2(x)}$ (but not given $mathbf{g_1(t)}$ and $mathbf{g_2(t)}$), can we say anything about the functional form that $mathbf{F(x)}$ must have for it to be transportable analytically?



There are a few constraints that $f(t,x)$ should adhere to:




  • The domain of $f(t,x)$ is $[0,infty)$, $(0,1]$ and its codomain is $[0,infty)$.

  • $f(t,1)=0$


  • $lim_{x downarrow 0} f(t,x) = +infty$.


  • $f(t,x)$ is a monotonically decreasing function of $x$.

  • $lim_{x downarrow 0} f^{(0,1)}(t,x) = -infty$

  • $lim_{x uparrow 1} f^{(0,1)}(t,x) = -infty$


In short, the inverse function of $f(t,x)$ w.r.t. its second argument could look like a Gaussian or a Lorentzian. In fact, the PDE considered here was derived from a much more horrible looking PDE formulated in terms of this inverse, as well as some other considerations.





1 They each depend on their own set of nonlinear evolution equations which I'm quite sure do not have a closed-form solution or approximation. To pull these equations into the context of this question would in all likelihood not improve it, I feel.










share|cite|improve this question















Note: this question was edited to reflect an evolving insight on part of the author. However, the original question still stands (although it has hopefully been formulated more clearly now).





For a physics problem that interests me, I am looking for a solution to the following linear homogeneous 1D transport equation with variable coefficient:



$$
f^{(1,0)}(t,x) - (g_1(t),h_1(x) + g_2(t),h_2(x));f^{(0,1)}(t,x) = 0
$$



In this PDE, all functions and variables are real-valued. $x$ has the meaning of a spatial coordinate and $t$ has the meaning of time. Superscripts between parentheses denote order of differentiation w.r.t. respective arguments. While the two functions of time $g_1(t)$ and $g_2(t)$ can best be considered unspecified1, both functions of space $h_1(x)$ and $h_2(x)$ are given by simple algebraic expressions:




  • $h_1(x)=+x(x^gamma-1)$


  • $h_2(x)=-x(x-1)$,


where $gamma$ is a positive real number. Since this is a transport equation, I have learned that the method of characteristics can be applied to transform a PDE into an ODE. As far as I have understood, I can write for the characteristic $X(t)$:



$$
frac{d}{dt};f(X(t),t) = 0
$$



$$
frac{dX}{dt} = - g_1(t),h_1(X(t)) - g_2(t),h_2(X(t))
$$



$$
f(t,x) = F(X^*(t,x))
$$



here, $X^*(t,x)$ is a solution to the characteristic equation above subject to initial conditions $X(0) = X^*(0,x) = x$. $F(x)$ is an unknown function.





Of course a transport equation will transport any signal we give it. Clearly though, an initial signal equal to e.g. sine of $x$ will not remain a sine of $x$ for long. My question is therefore: given $mathbf{h_1(x)}$ and $mathbf{h_2(x)}$ (but not given $mathbf{g_1(t)}$ and $mathbf{g_2(t)}$), can we say anything about the functional form that $mathbf{F(x)}$ must have for it to be transportable analytically?



There are a few constraints that $f(t,x)$ should adhere to:




  • The domain of $f(t,x)$ is $[0,infty)$, $(0,1]$ and its codomain is $[0,infty)$.

  • $f(t,1)=0$


  • $lim_{x downarrow 0} f(t,x) = +infty$.


  • $f(t,x)$ is a monotonically decreasing function of $x$.

  • $lim_{x downarrow 0} f^{(0,1)}(t,x) = -infty$

  • $lim_{x uparrow 1} f^{(0,1)}(t,x) = -infty$


In short, the inverse function of $f(t,x)$ w.r.t. its second argument could look like a Gaussian or a Lorentzian. In fact, the PDE considered here was derived from a much more horrible looking PDE formulated in terms of this inverse, as well as some other considerations.





1 They each depend on their own set of nonlinear evolution equations which I'm quite sure do not have a closed-form solution or approximation. To pull these equations into the context of this question would in all likelihood not improve it, I feel.







calculus real-analysis differential-equations multivariable-calculus pde






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edited Dec 4 '18 at 10:53







Casper Pranger

















asked Nov 30 '18 at 11:17









Casper PrangerCasper Pranger

12




12












  • What's non-linear about this equation? You have $frac{partial f}{partial t} = a(x,t)frac{partial f}{partial x}$
    – Dylan
    Nov 30 '18 at 12:17












  • You are correct. I have changed the title and text accordingly.
    – Casper Pranger
    Nov 30 '18 at 13:28










  • Thanks to the remark of @Dylan that this is not a non-linear equation at all, I now realize that its classification would be 'one-dimensional transport equation with variable coefficient'. I am currently looking into applying the method of characteristics, though so far I can not say that it is very helpful to me. If I find anything concrete, I will be sure to update the question.
    – Casper Pranger
    Nov 30 '18 at 17:40


















  • What's non-linear about this equation? You have $frac{partial f}{partial t} = a(x,t)frac{partial f}{partial x}$
    – Dylan
    Nov 30 '18 at 12:17












  • You are correct. I have changed the title and text accordingly.
    – Casper Pranger
    Nov 30 '18 at 13:28










  • Thanks to the remark of @Dylan that this is not a non-linear equation at all, I now realize that its classification would be 'one-dimensional transport equation with variable coefficient'. I am currently looking into applying the method of characteristics, though so far I can not say that it is very helpful to me. If I find anything concrete, I will be sure to update the question.
    – Casper Pranger
    Nov 30 '18 at 17:40
















What's non-linear about this equation? You have $frac{partial f}{partial t} = a(x,t)frac{partial f}{partial x}$
– Dylan
Nov 30 '18 at 12:17






What's non-linear about this equation? You have $frac{partial f}{partial t} = a(x,t)frac{partial f}{partial x}$
– Dylan
Nov 30 '18 at 12:17














You are correct. I have changed the title and text accordingly.
– Casper Pranger
Nov 30 '18 at 13:28




You are correct. I have changed the title and text accordingly.
– Casper Pranger
Nov 30 '18 at 13:28












Thanks to the remark of @Dylan that this is not a non-linear equation at all, I now realize that its classification would be 'one-dimensional transport equation with variable coefficient'. I am currently looking into applying the method of characteristics, though so far I can not say that it is very helpful to me. If I find anything concrete, I will be sure to update the question.
– Casper Pranger
Nov 30 '18 at 17:40




Thanks to the remark of @Dylan that this is not a non-linear equation at all, I now realize that its classification would be 'one-dimensional transport equation with variable coefficient'. I am currently looking into applying the method of characteristics, though so far I can not say that it is very helpful to me. If I find anything concrete, I will be sure to update the question.
– Casper Pranger
Nov 30 '18 at 17:40










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