The Fourier Transform of a piecewise smooth function f(x), for which f(x)|dx exists and is finite, is given by 1 F(k) =F[f(x)] = = 2πT √2 f(x)e-ik² dr. In particular we have F(ak) = ¼Ƒ[ƒ(4)] and F[e¯³] = e. Consider the differential equation u₁ = tuxx -xx<∞, subject to initial condition u(x, 0) = e− Defining (k,t) = F[u(x, t)] (1+t2)k2\ û(k,t) = exp 2 (4) (5) Show that the solution of equation (4) satisfying the initial condition (5) has the form 1 u(x,t) = = exp 1+t2 (-2 x² 2(1+t2)
The Fourier Transform of a piecewise smooth function f(x), for which f(x)|dx exists and is finite, is given by 1 F(k) =F[f(x)] = = 2πT √2 f(x)e-ik² dr. In particular we have F(ak) = ¼Ƒ[ƒ(4)] and F[e¯³] = e. Consider the differential equation u₁ = tuxx -xx<∞, subject to initial condition u(x, 0) = e− Defining (k,t) = F[u(x, t)] (1+t2)k2\ û(k,t) = exp 2 (4) (5) Show that the solution of equation (4) satisfying the initial condition (5) has the form 1 u(x,t) = = exp 1+t2 (-2 x² 2(1+t2)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The Fourier Transform of a piecewise smooth function f(x), for which |f(x)|dx
exists and is finite, is given by
1
F(k) =F[f(x)] =
=
√2 f(x)e-ik² dr.
2πT
In particular we have F(ak) = ¼Ƒ[ƒ(4)] and F[e¯½³] = e.
Consider the differential equation
u₁ = tuxx
-8<x<∞,
subject to initial condition
u(x, 0) = e−².
Defining (k,t) = F[u(x, t)]
(1+t2)k2\
û(k,t) = exp
2
(4)
(5)
Show that the solution of equation (4) satisfying the initial condition (5) has
the form
1
u(x,t) =
=
exp
1+t2
(-2
x²
2(1+t2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7560de44-7bbe-4fed-b63a-532eb75ed369%2Fd89c6b81-589d-4f5b-b264-f5e76ca187ae%2Fve2l7a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The Fourier Transform of a piecewise smooth function f(x), for which |f(x)|dx
exists and is finite, is given by
1
F(k) =F[f(x)] =
=
√2 f(x)e-ik² dr.
2πT
In particular we have F(ak) = ¼Ƒ[ƒ(4)] and F[e¯½³] = e.
Consider the differential equation
u₁ = tuxx
-8<x<∞,
subject to initial condition
u(x, 0) = e−².
Defining (k,t) = F[u(x, t)]
(1+t2)k2\
û(k,t) = exp
2
(4)
(5)
Show that the solution of equation (4) satisfying the initial condition (5) has
the form
1
u(x,t) =
=
exp
1+t2
(-2
x²
2(1+t2)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
