Upcoming SlideShare
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×

# Saving this for later?

### Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime - even offline.

Standard text messaging rates apply

# Application of fourier series to differential equations

878
views

Published on

Application of fourier series to differential equations

Application of fourier series to differential equations

Published in: Education, Technology

1 Like
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total Views
878
On Slideshare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
18
0
Likes
1
Embeds 0
No embeds

No notes for slide

### Transcript

• 1. TARUN GEHLOT (B.E, CIVIL HONORS) Application of Fourier Series to Differential Equations Since the beginning Fourier himself was interested to find a powerful tool to be used in solving differential equations. Therefore, it is of no surprise that we discuss in this page, the application of Fourier series differential equations. We will only discuss the equations of the form y(n) + an-1y(n-1) + ........+ a1y' + a0 y = f(x), where f(x) is a -periodic function. Note that we will need the complex form of Fourier series of a periodic function. Let us define this object first: Definition. Let f(x) be -periodic. The complex Fourier series of f(x) is where We will use the notation If you wonder about the existence of a relationship between the real Fourier coefficients and the complex ones, the next theorem answers that worry. Theoreme. Let f(x) be -periodic. Consider the real Fourier coefficients and of f(x), as well as the complex Fourier coefficients . We have
• 2. TARUN GEHLOT (B.E, CIVIL HONORS) The proof is based on Euler's formula for the complex exponential function. Remark. When f(x) is 2L-periodic, then the complex Fourier series will be defined as before where for any . Example. Let f(x) = x, for and f(x+2) = f(x). Find its complex Fourier coefficients . Answer. We have d0 = 0 and Easy calculations give Since , we get . Consequently
• 3. TARUN GEHLOT (B.E, CIVIL HONORS) Back to our original problem. In order to apply the Fourier technique to differential equations, we will need to have a result linking the complex coefficients of a function with its derivative. We have: Theorem. Let f(x) be 2L-periodic. Assume that f(x) is differentiable. If then Example. Find the periodic solutions of the differential equation y' + 2y = f(x), where f(x) is a -periodic function. Answer. Set Let y be any -periodic solution of the differential equation. Assume Then, from the differential equation, we get Hence
• 4. TARUN GEHLOT (B.E, CIVIL HONORS) Therefore, we have Example. Find the periodic solutions of the differential equation Answer. Because we get with Let y be a periodic solution of the differential equation. If then . Hence
• 5. TARUN GEHLOT (B.E, CIVIL HONORS) Therefore, the differential equation has only one periodic solution The most important result may be stated as: Theoreme. Consider the differential equation where f(x) is a 2c-periodic function. Assume for . Then the differential equations has one 2c-periodic solution given by where