1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

128 Chapter 1 Fourier Series and Integrals 22.Find the Fourier sine and cosine integral representations of the function given by ...

Miscellaneous Exercises 129 a. 2 +4sin( 50 x)−12 cos( 41 x); c.sin( 4 x+ 2 ); e.cos^3 (x); b.sin^2 ( 5 x); d. sin( 3 x)cos( 5 x) ...

130 Chapter 1 Fourier Series and Integrals c.sketch the even periodic extension of the given function for at least two periods. ...

Miscellaneous Exercises 131 & 52.f(x)=    0 , 0 <x<a 2 , x=−a,a 2 ,a, 1 , a 2 <x<a. 53–58.For each of these ...

132 Chapter 1 Fourier Series and Integrals 66.In analogy to Lemma 2 of Section 7, prove that N∑− 1 n= 0 sin (( n+ 1 2 ) y ) = si ...

Miscellaneous Exercises 133 a. |x|=^1 2 −^4 π^2 ∑∞ k= 0 1 ( 2 k+ 1 )^2 cos ( ( 2 k+ 1 )πx ) , − 1 <x<1, π^2 8 = 1 +^1 9 +^ ...

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The Heat Equation CHAPTER 2 2.1 Derivation and Boundary Conditions As the first example of the derivation of a partial different ...

136 Chapter 2 The Heat Equation Figure 1 Rodofheat-conductingmaterial. Figure 2 Slice cut from rod. The rate of heat storage in ...

Chapter 2 The Heat Equation 137 should be recognized as a difference quotient. If we allow xto decrease, this quotient becomes, ...

138 Chapter 2 The Heat Equation c ρκk=ρκc Material (gcal◦C)(cmg 3 )(scmcal◦C)(cms^2 ) Aluminum 0. 21 2. 70. 48 0. 83 Copper 0. 0 ...

Chapter 2 The Heat Equation 139 whereαis a function of time. Of course, the case of a constant function is included here. This t ...

140 Chapter 2 The Heat Equation u( 0 ,t)=u(a,t), t> 0 , (11) ∂u ∂x(^0 ,t)= ∂u ∂x(a,t), t>^0 , (12) both of mixed type. Man ...

Chapter 2 The Heat Equation 141 When we rearrange Eq. (17) and take the limit as xapproaches 0, it becomes −∂q ∂x +g=∂u ∂t . (18 ...

142 Chapter 2 The Heat Equation EXERCISES 1.Give a physical interpretation for the problem in Eqs. (13)–(16). 2.Verify that the ...

2.2 Steady-State Temperatures 143 Under what conditions might the second factor on the right be taken ap- proximately constant? ...

144 Chapter 2 The Heat Equation Example. For the preceding problem,v(x)should be the solution to the problem d^2 v dx^2 =^0 ,^0 ...

2.2 Steady-State Temperatures 145 When the rule given here is applied to this problem, we are led to the following equations: d^ ...

146 Chapter 2 The Heat Equation It is easy to see thatv(x)=T(any constant) is a solution to this problem. However, there is no i ...

2.2 Steady-State Temperatures 147 Now we collect and simplify these transformations of Eqs. (1)–(4) to get an initial value–boun ...