- #1
chwala
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- Homework Statement
- Kindly see the attachment below
- Relevant Equations
- D'Alembert approach (change of variables )
I am refreshing on the pde's, and i am trying to understand how the textbook was addressing change of variables, i find it a bit confusing. I will share the textbook approach, then later share my own understanding on change of variables approach. Here is the textbook approach;
My approach on this change of variables would look like this,
Let the pde be of the form;
##au_{xx}+bu_{xt}+cu_{tt}+du_x+eu_t+fu=0##
Consider the pde; ##u_{tt}= c^2u_{xx}## then we may use the change of variables as indicated by literature into working out the solution...
Now letting, ##ξ = x +ct## and ##η= x-ct##, then it follows that,
##u_x##=## u_ξ⋅ξ_x + u_η⋅η_x## =##u_ξ + η_x##
##u_{xx}## = ##u_x⋅u_ξ⋅ξ_x + u_x⋅u_η⋅η_x## = ##u_x⋅u_ξ+u_x⋅u_η## = ##[u_ξ⋅ξ_x + u_η⋅η_x]⋅u_ξ +[u_ξ⋅ξ_x + u_η⋅η_x]⋅u_η## = ##u_{ξξ} + 2u_{ξη} + u_{ηη}##
Similarly, i can show that,
##u_t## =## cu_ξ - cu_η ## ... Note that ##[u_t## =##u_ξ ⋅ξ_t + u_η ⋅η_t]##
##u_{tt}##= ##u_t⋅u_ξ⋅ξ_t +u_t ⋅u_η⋅ η_t##=... ##c^2u_{ξξ}-2c^2u_ξ⋅u_η +c^2u_{ηη}##
also,
##u_{xt}##= ##u_x⋅u_ξ⋅ξ_t + u_x⋅u_η⋅η_t##
My point is that, since i understand this quite well then i should not bother with the textbook approach as both ways would work towards the same solution...right?
My approach on this change of variables would look like this,
Let the pde be of the form;
##au_{xx}+bu_{xt}+cu_{tt}+du_x+eu_t+fu=0##
Consider the pde; ##u_{tt}= c^2u_{xx}## then we may use the change of variables as indicated by literature into working out the solution...
Now letting, ##ξ = x +ct## and ##η= x-ct##, then it follows that,
##u_x##=## u_ξ⋅ξ_x + u_η⋅η_x## =##u_ξ + η_x##
##u_{xx}## = ##u_x⋅u_ξ⋅ξ_x + u_x⋅u_η⋅η_x## = ##u_x⋅u_ξ+u_x⋅u_η## = ##[u_ξ⋅ξ_x + u_η⋅η_x]⋅u_ξ +[u_ξ⋅ξ_x + u_η⋅η_x]⋅u_η## = ##u_{ξξ} + 2u_{ξη} + u_{ηη}##
Similarly, i can show that,
##u_t## =## cu_ξ - cu_η ## ... Note that ##[u_t## =##u_ξ ⋅ξ_t + u_η ⋅η_t]##
##u_{tt}##= ##u_t⋅u_ξ⋅ξ_t +u_t ⋅u_η⋅ η_t##=... ##c^2u_{ξξ}-2c^2u_ξ⋅u_η +c^2u_{ηη}##
also,
##u_{xt}##= ##u_x⋅u_ξ⋅ξ_t + u_x⋅u_η⋅η_t##
My point is that, since i understand this quite well then i should not bother with the textbook approach as both ways would work towards the same solution...right?
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