How to Solve a Second Order Partial Differential Equation
Check whether it is hyperbolic, elliptic or parabolic., Calculate two quantities., Write out the two equations below., Call your constants η{\displaystyle \eta } and ν{\displaystyle \nu }., Calculate the partial derivatives in terms of these new...
Step-by-Step Guide
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Step 1: Check whether it is hyperbolic
To do this, calculate the discriminant D=B2−AC.{\displaystyle D=B^{2}-AC.} If this is positive, the PDE is hyperbolic.
Negative means it is elliptic, and if it equals zero, it is parabolic. , Call them λ+{\displaystyle \lambda _{+}} and λ−{\displaystyle \lambda _{-}}. λ±=B±DA{\displaystyle \lambda _{\pm }={\frac {B\pm {\sqrt {D}}}{A}}} , Solve both of these, and put your constants of integration onto the left and everything else on the right. dy±dx=λ±{\displaystyle {\frac {{\mathrm {d} }y_{\pm }}{{\mathrm {d} }x}}=\lambda _{\pm }} , You should now have two equations of the form η=f(x,y){\displaystyle \eta =f(x,y)} and ν=g(x,y).{\displaystyle \nu =g(x,y).} , Using the chain rule ux=uηηx+uννx,{\displaystyle u_{x}=u_{\eta }\eta _{x}+u_{\nu }\nu _{x},} you can plug in the values for ηx{\displaystyle \eta _{x}} and νx{\displaystyle \nu _{x}} by differentiating your equations for η{\displaystyle \eta } and ν.{\displaystyle \nu .} Do the same thing for uy{\displaystyle u_{y}} where uy=uηηy+uννy.{\displaystyle u_{y}=u_{\eta }\eta _{y}+u_{\nu }\nu _{y}.} , For example, the result for uxx{\displaystyle u_{xx}} is given below.
In the same way, you can find uxy{\displaystyle u_{xy}} and uyy.{\displaystyle u_{yy}.} Don't forget to plug in all values you can calculate from the equations for η{\displaystyle \eta } and ν.{\displaystyle \nu .} uxx=(uηηηx+uηννx)ηx+(uηνηx+uνννx)νx{\displaystyle u_{xx}=(u_{\eta \eta }\eta _{x}+u_{\eta \nu }\nu _{x})\eta _{x}+(u_{\eta \nu }\eta _{x}+u_{\nu \nu }\nu _{x})\nu _{x}} , Your equation is now ready to be solved given certain boundary conditions, otherwise, this is as far as you can go. -
Step 2: elliptic or parabolic.
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Step 3: Calculate two quantities.
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Step 4: Write out the two equations below.
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Step 5: Call your constants η{\displaystyle \eta } and ν{\displaystyle \nu }.
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Step 6: Calculate the partial derivatives in terms of these new coordinates.
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Step 7: Calculate the second partial derivatives by differentiating these again to find uxx
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Step 8: {\displaystyle u_{xx}
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Step 9: {\displaystyle u_{xy}
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Step 10: } and uyy{\displaystyle u_{yy}}.
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Step 11: Plug your values for uxx
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Step 12: {\displaystyle u_{xx}
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Step 13: {\displaystyle u_{xy}
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Step 14: {\displaystyle u_{yy}
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Step 15: {\displaystyle u_{x}
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Step 16: } and uy{\displaystyle u_{y}} back into the original equation.
Detailed Guide
To do this, calculate the discriminant D=B2−AC.{\displaystyle D=B^{2}-AC.} If this is positive, the PDE is hyperbolic.
Negative means it is elliptic, and if it equals zero, it is parabolic. , Call them λ+{\displaystyle \lambda _{+}} and λ−{\displaystyle \lambda _{-}}. λ±=B±DA{\displaystyle \lambda _{\pm }={\frac {B\pm {\sqrt {D}}}{A}}} , Solve both of these, and put your constants of integration onto the left and everything else on the right. dy±dx=λ±{\displaystyle {\frac {{\mathrm {d} }y_{\pm }}{{\mathrm {d} }x}}=\lambda _{\pm }} , You should now have two equations of the form η=f(x,y){\displaystyle \eta =f(x,y)} and ν=g(x,y).{\displaystyle \nu =g(x,y).} , Using the chain rule ux=uηηx+uννx,{\displaystyle u_{x}=u_{\eta }\eta _{x}+u_{\nu }\nu _{x},} you can plug in the values for ηx{\displaystyle \eta _{x}} and νx{\displaystyle \nu _{x}} by differentiating your equations for η{\displaystyle \eta } and ν.{\displaystyle \nu .} Do the same thing for uy{\displaystyle u_{y}} where uy=uηηy+uννy.{\displaystyle u_{y}=u_{\eta }\eta _{y}+u_{\nu }\nu _{y}.} , For example, the result for uxx{\displaystyle u_{xx}} is given below.
In the same way, you can find uxy{\displaystyle u_{xy}} and uyy.{\displaystyle u_{yy}.} Don't forget to plug in all values you can calculate from the equations for η{\displaystyle \eta } and ν.{\displaystyle \nu .} uxx=(uηηηx+uηννx)ηx+(uηνηx+uνννx)νx{\displaystyle u_{xx}=(u_{\eta \eta }\eta _{x}+u_{\eta \nu }\nu _{x})\eta _{x}+(u_{\eta \nu }\eta _{x}+u_{\nu \nu }\nu _{x})\nu _{x}} , Your equation is now ready to be solved given certain boundary conditions, otherwise, this is as far as you can go.
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Adam Richardson
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