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If we use the Fourier transform  to solve the partial differential equation   in the half-plane {(x, y) : -∞ < x < ∞, 0 < y < ∞} the Fourier modes ϕk(y) depend on y as yα and yβ. The values of α and β are  
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If we use the Fourier transformto solve the partial differential equat...
The Fourier transform, 
CALCULATION:
We have;

Now take the derivative of ϕ(x, y) we have;

Again derivative we have;

Now putting these values in equation 1) we have;

Now, by applying Cauchy's differential equation we have;

Again solving we have;
The roots are written as;

Hence option 3) is the correct answer.
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Most Upvoted Answer
If we use the Fourier transformto solve the partial differential equat...
The Fourier transform, 
CALCULATION:
We have;

Now take the derivative of ϕ(x, y) we have;

Again derivative we have;

Now putting these values in equation 1) we have;

Now, by applying Cauchy's differential equation we have;

Again solving we have;
The roots are written as;

Hence option 3) is the correct answer.
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If we use the Fourier transformto solve the partial differential equationin the half-plane {(x, y) : -∞ < x < ∞, 0 < y < ∞} the Fourier modes k(y) depend on y as yα and yβ. The values of α and β are a)b)c)d)Correct answer is option 'C'. Can you explain this answer?
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