Solving Differential Eqs with Critical Point: Initial Conditions & Solutions

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SUMMARY

This discussion addresses the complexities of solving a system of differential equations with specific initial conditions, particularly focusing on the equations involving density ρ and velocity u as functions of radius r. The user highlights the existence of multiple solutions for the same boundary conditions, specifically referencing the constant \(\dot{M}=4\pi r^2u\rho=const\). The user attempts to solve the system numerically using Mathematica's NDSolve function but encounters divergence issues at the critical point \(r_c\). The equations presented include \(\frac{\rho'}{\rho}+\frac{u'}{u}+\frac{2}{r}=0\) and \(uu'+a^2(r)\frac{\rho'}{\rho}+\frac{\alpha}{r^2}=0\), with \(a(r)\) defined as \(a_0\left(\frac{\rho(r)}{\rho_0}\right)^{(\Gamma-1)/2}\).

PREREQUISITES
  • Understanding of differential equations and their boundary conditions.
  • Familiarity with numerical methods for solving differential equations.
  • Proficiency in using Mathematica, specifically the NDSolve function.
  • Knowledge of physical concepts related to fluid dynamics, particularly in astrophysical contexts.
NEXT STEPS
  • Research the implications of varying the constant \(\dot{M}\) on solution classes in differential equations.
  • Study numerical methods for solving differential equations, focusing on stability and convergence issues.
  • Explore advanced features of Mathematica's NDSolve, including parameter sensitivity analysis.
  • Investigate the differences between boundary conditions and initial conditions in the context of differential equations.
USEFUL FOR

Researchers, physicists, and mathematicians dealing with differential equations in fluid dynamics, particularly those interested in numerical solutions and boundary value problems.

Vrbic
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Hello,
I have two different discrepancies to this system:
a) How and when is possible to have more solution of differential eq. or their system for same initial problem? For example this is happening in following system. It is written about this system:
"Different value of constant \dot{M}=4\pi r^2u\rho=const. lead to physically distinct classes of solution for the same boundary condition at infinity". Can anybody explain it to me?

\frac{\rho'}{\rho}+\frac{u'}{u}+\frac{2}{r}=0
uu'+a^2(r)\frac{\rho'}{\rho}+\frac{\alpha}{r^2}=0, where a(r)=a_0\big(\frac{\rho(r)}{\rho_0}\big)^{(\Gamma-1)/2}, \rho(r) and u(r) are function of r and \Gamma, a_0, \rho_0, \alpha are constant.

b) How to solve this system? I am quit sure that is possible just numerically and I tried. I expressed this equation in this way:
u'=\frac{D_1}{D}
u'=\frac{D_2}{D}, where D_1=\frac{2a^2/r-\alpha/r^2}{\rho}, D_2=\frac{2u^2/r-\alpha/r^2}{u} and D=\frac{u^2-a^2}{\rho u}.
No I see if I want smooth solution I need D_1=D_2=D=0 in same critical point r_c. These are two boundary condition (Im not sure what difference is between boundary and initial condition?). I find out r_c and u_c, value of u(r_c)=u_c. So I have to condition a(r_c)=u(r_c) and u(r_c)=u_c Am I right till this time?
Now is coming my question:
Im not much familiar with numerical methods or numerical solving so I used Mathematica soft, concretely function NDSolve with conditions mentioned above. And Mathematica said "no". By "no" I mean it diverges in r_c. What could be wrong?
 
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Vrbic said:
Hello,
I have two different discrepancies to this system:
a) How and when is possible to have more solution of differential eq. or their system for same initial problem? For example this is happening in following system. It is written about this system:
"Different value of constant \dot{M}=4\pi r^2u\rho=const. lead to physically distinct classes of solution for the same boundary condition at infinity". Can anybody explain it to me?

\frac{\rho'}{\rho}+\frac{u'}{u}+\frac{2}{r}=0
uu'+a^2(r)\frac{\rho'}{\rho}+\frac{\alpha}{r^2}=0, where a(r)=a_0\big(\frac{\rho(r)}{\rho_0}\big)^{(\Gamma-1)/2}, \rho(r) and u(r) are function of r and \Gamma, a_0, \rho_0, \alpha are constant.

b) How to solve this system? I am quit sure that is possible just numerically and I tried. I expressed this equation in this way:
u'=\frac{D_1}{D}
u'=\frac{D_2}{D}, where D_1=\frac{2a^2/r-\alpha/r^2}{\rho}, D_2=\frac{2u^2/r-\alpha/r^2}{u} and D=\frac{u^2-a^2}{\rho u}.
No I see if I want smooth solution I need D_1=D_2=D=0 in same critical point r_c. These are two boundary condition (Im not sure what difference is between boundary and initial condition?). I find out r_c and u_c, value of u(r_c)=u_c. So I have to condition a(r_c)=u(r_c) and u(r_c)=u_c Am I right till this time?
Now is coming my question:
Im not much familiar with numerical methods or numerical solving so I used Mathematica soft, concretely function NDSolve with conditions mentioned above. And Mathematica said "no". By "no" I mean it diverges in r_c. What could be wrong?
Mistake in second equation in section b) so again both:
u'=\frac{D_1}{D}
\rho'=\frac{D_2}{D},
 

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