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Consider the diffusion equation with convection on the real line. მu. at = K J²u მ2 მu. + C- მე u(0,x) = f(x) (∞0>x>∞0-) a) Take the Fourier Transform of the PDE and initial data with respect tox and solve for u(t, w). b) Using the fact that the inverse Fourier Transform of a Gaussian is F(w) = e-aw² f(x) = -x²/(4α) (a > 0) write the solution u(t, x) to the convection equation as a convolution of the initial data f(x) with an appropriate function. (HINT: You should get a term like eicut in u. This should simply shift x to x + ct in the final answer!)

Consider the diffusion equation with convection on the real line. მu. at = K J²u მ2 მu. + C- მე u(0,x) = f(x) (∞0>x>∞0-) a) Take the Fourier Transform of the PDE and initial data with respect tox and solve for u(t, w). b) Using the fact that the inverse Fourier Transform of a Gaussian is F(w) = e-aw² f(x) = -x²/(4α) (a > 0) write the solution u(t, x) to the convection equation as a convolution of the initial data f(x) with an appropriate function. (HINT: You should get a term like eicut in u. This should simply shift x to x + ct in the final answer!)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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Consider the diffusion equation with convection on the real line. მu. at = K J²u მ2 მu. + C- მე u(0,x) = f(x) (∞0>x>∞0-) a) Take the Fourier Transform of the PDE and initial data with respect tox and solve for u(t, w). b) Using the fact that the inverse Fourier Transform of a Gaussian is F(w) = e-aw² f(x) = -x²/(4α) (a > 0) write the solution u(t, x) to the convection equation as a convolution of the initial data f(x) with an appropriate function. (HINT: You should get a term like eicut in u. This should simply shift x to x + ct in the final answer!)
Transcribed Image Text:Consider the diffusion equation with convection on the real line. მu. at = K J²u მ2 მu. + C- მე u(0,x) = f(x) (∞0>x>∞0-) a) Take the Fourier Transform of the PDE and initial data with respect tox and solve for u(t, w). b) Using the fact that the inverse Fourier Transform of a Gaussian is F(w) = e-aw² f(x) = -x²/(4α) (a > 0) write the solution u(t, x) to the convection equation as a convolution of the initial data f(x) with an appropriate function. (HINT: You should get a term like eicut in u. This should simply shift x to x + ct in the final answer!)
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