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How to handle a differential equation whose RHS contains the derivative of the Dirac delta function?

Solution of differential equation with Dirac DeltaGetting 0 solving Schrodinger equation with Dirac delta by Fourier transformIntegral of delta dirac functionDiffusion equation involving dirac delta termFirst order differential equation with a derivative delta functionAbsolute value: First Derivative Heaviside Function + Second Derivative Dirac Delta Function DistributionDifferential Equation with Delta DiracSolving differential equation with the Dirac Delta FunctionVerifying the delta function satisfies Poisson's EquationHow to solve the differential equation with delta function on right hand side

1

$begingroup$

I have a differential equation of the following form.

beginalign
beginsplit
fracmathrmd^4psi(eta)mathrmdeta^4-beta^4psi(eta)=psi'(zeta),delta'(eta-zeta)
endsplit
endalign

How to handle the derivative of the Dirac delta function on the right-hand side? I am trying to find a closed form solution using Green's functions.

Suppose if a differential equation contains both Dirac delta function and its derivative, can I linearly superimpose the solution of the differential equation of just Dirac delta function plus the solution of the differential equation with derivative of Dirac delta function?

beginalign
beginsplit
fracmathrmd^4psi(eta)mathrmdeta^4-beta^4psi(eta)=psi(zeta),delta(eta-zeta)+psi'(zeta),delta'(eta-zeta)
endsplit
endalign

ordinary-differential-equations closed-form dirac-delta

share|cite|improve this question

edited Mar 16 at 11:43

Rodrigo de Azevedo

13.2k41960

asked Mar 16 at 11:18

musimathicsmusimathics

1418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually ,my the goal is to extract the solution using Green's function approach
  $endgroup$
  – musimathics
  Mar 16 at 11:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually I was editing the question, It may be happened accddentally. Please suggest , so that I can make changes to the question
  $endgroup$
  – musimathics
  Mar 16 at 11:33
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Assuming zero initial conditions (all four of them), the Laplace transform is $$Psi (s) = psi'(zeta) left(fracss^4 - beta^4right)$$ From here, one can use partial fraction expansion to conclude what the form of the solution is.
  $endgroup$
  – Rodrigo de Azevedo
  Mar 16 at 11:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This is using Laplace transforms. Is there any ways that I can fit a Green's function to this DE.
  $endgroup$
  – musimathics
  Mar 16 at 11:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am not familiar with Green's functions. At least, not that I know of.
  $endgroup$
  – Rodrigo de Azevedo
  Mar 16 at 11:50
  

  
  
  
  

add a comment | 

1

$begingroup$

I have a differential equation of the following form.

beginalign
beginsplit
fracmathrmd^4psi(eta)mathrmdeta^4-beta^4psi(eta)=psi'(zeta),delta'(eta-zeta)
endsplit
endalign

How to handle the derivative of the Dirac delta function on the right-hand side? I am trying to find a closed form solution using Green's functions.

Suppose if a differential equation contains both Dirac delta function and its derivative, can I linearly superimpose the solution of the differential equation of just Dirac delta function plus the solution of the differential equation with derivative of Dirac delta function?

beginalign
beginsplit
fracmathrmd^4psi(eta)mathrmdeta^4-beta^4psi(eta)=psi(zeta),delta(eta-zeta)+psi'(zeta),delta'(eta-zeta)
endsplit
endalign

ordinary-differential-equations closed-form dirac-delta

share|cite|improve this question

edited Mar 16 at 11:43

Rodrigo de Azevedo

13.2k41960

asked Mar 16 at 11:18

musimathicsmusimathics

1418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually ,my the goal is to extract the solution using Green's function approach
  $endgroup$
  – musimathics
  Mar 16 at 11:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually I was editing the question, It may be happened accddentally. Please suggest , so that I can make changes to the question
  $endgroup$
  – musimathics
  Mar 16 at 11:33
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Assuming zero initial conditions (all four of them), the Laplace transform is $$Psi (s) = psi'(zeta) left(fracss^4 - beta^4right)$$ From here, one can use partial fraction expansion to conclude what the form of the solution is.
  $endgroup$
  – Rodrigo de Azevedo
  Mar 16 at 11:47
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This is using Laplace transforms. Is there any ways that I can fit a Green's function to this DE.
  $endgroup$
  – musimathics
  Mar 16 at 11:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am not familiar with Green's functions. At least, not that I know of.
  $endgroup$
  – Rodrigo de Azevedo
  Mar 16 at 11:50
  

  
  
  
  

add a comment | 

$begingroup$

I have a differential equation of the following form.

beginalign
beginsplit
fracmathrmd^4psi(eta)mathrmdeta^4-beta^4psi(eta)=psi'(zeta),delta'(eta-zeta)
endsplit
endalign

How to handle the derivative of the Dirac delta function on the right-hand side? I am trying to find a closed form solution using Green's functions.

Suppose if a differential equation contains both Dirac delta function and its derivative, can I linearly superimpose the solution of the differential equation of just Dirac delta function plus the solution of the differential equation with derivative of Dirac delta function?

beginalign
beginsplit
fracmathrmd^4psi(eta)mathrmdeta^4-beta^4psi(eta)=psi(zeta),delta(eta-zeta)+psi'(zeta),delta'(eta-zeta)
endsplit
endalign

ordinary-differential-equations closed-form dirac-delta

share|cite|improve this question

edited Mar 16 at 11:43

Rodrigo de Azevedo

13.2k41960

asked Mar 16 at 11:18

musimathicsmusimathics

1418

$endgroup$

share|cite|improve this question

edited Mar 16 at 11:43

Rodrigo de Azevedo

13.2k41960

asked Mar 16 at 11:18

musimathicsmusimathics

1418

share|cite|improve this question

edited Mar 16 at 11:43

Rodrigo de Azevedo

13.2k41960

asked Mar 16 at 11:18

musimathicsmusimathics

1418

edited Mar 16 at 11:43

Rodrigo de Azevedo

13.2k41960

edited Mar 16 at 11:43

Rodrigo de Azevedo

13.2k41960

asked Mar 16 at 11:18

musimathicsmusimathics

1418

asked Mar 16 at 11:18

musimathicsmusimathics

1418