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## FOURIER ANALYSIS

by: Cassidy Grimes

10

0

2

# FOURIER ANALYSIS MATH 750

Cassidy Grimes

GPA 3.51

Staff

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COURSE
PROF.
Staff
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Class Notes
PAGES
2
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KARMA
25 ?

## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 750 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/229556/math-750-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15
Math 750 Introduction Outline of Fourier7s method We show that under certain assumptions that heat is governed by a partial differential equation on a domain D 8 87 a An 1 and satis es the initial boundary value conditions iix7 0 m E D U 8D 2 iix7 t gz7 t7 m E 3Dt gt 0 Here iim7 t is the temperature of the body at location z E D at time t Fourier formulated the empirical law for heat ow ie Fourier s law qx7 t 7K Va 3 which states that the time rate of change of heat is proportional to the spatial rate of change of the temperature with heat 7 owing7 from warmer regions to cooler regions K is called the thermal conductivity and depends upon the material properties of the body D For convenience we assume that K is constant Rate of heat passing through 3D The amount of heat leaving a small volume D C D through its boundary is equal to the surface integral 8DqndA 4 Applying the divergence theorem to this equation7 we get qndAVqu 5 3D D and so substituting for q from Fourier s law 37 we obtain that the heat ow the small volume D gains hence the change in sign is equal to KAu dV 6 D This is under the assumption that there are no sources or sinks of heat inside D Rate of change of heat of D On the other hand the total heat in the small volume D is equal to the volume integral A Upu dV 7 where a is the material speci c heat heat gain per unit mass per unit change in temperature and p is the material mass density mass per unit volume Therefore the time rate of change of the heat equals 1 8n EltDapudVgtDapEdV 8 where in this equation we have used Leibnitz s rule for differentiation of integrals and for convenience have assumed that a and p are constant Heat Balance We equate the two rates in expressions 6 and 8 ie the heat balance subtract and divide by the volume of D to obtain mD gpgijizmug dV0 9 But D is an arbitrary region inside D so we may vary D about any point in D letting its volume tend to zero in 9 and apply the Lebesgue Differentiation Theorem to formally obtain that 8 87 7 r Au 0 LE 10 where a g If these quantities are continuous then the PDE for u holds on all of D Solution in 1 D We use the elementary separation of variable technique assume a solution of the form Una 75 XW Tt7 11 substitute into the PDE divide by U to obtain T X i it 12 T H X The left hand side is only a function oft while the right hand side is only a function of z so they are both constant and coupled through this relationship X Y 7A 13 and T 7 7rlt 14 For illustration we take the 1 D domain D to be the interval 07139 and take zero boundary conditions in By considering all possible cases in 15 we see that the eigenvalues 0 lt 712 n 6 IV is necessary for any nonzero solution to the PDE and Xnm bn sinnz 15 If we take the domain as 77172 7r2 for even functions 1 then Xnz an cosnm Substituting this value of into the coupled equation for the temporal component we get Tnt eXp7n2m 16 and so the tensor product solution is Unmt bn sinnz exp7n2m 17 Superimposing the solutions Un and using the linearity of the initial boundary value problem we see that a solution is of the form umt anexp7n2mf sinnx 18 n1 Setting t O a formal solution is found by solving for coef cients bn in the representation 00 m 1 sinnx 19 n1 Now the analysis takes center stage When and in what sense does this sum converge that is when does equality hold and for what 1 is the representation 19 valid Once this is addressed whenwhere does the solution 18 converge and solve the PDE ie is this really a solution in D when it does converge Is this solution unique As if L O in what sense does u t converge to f

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