1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

288 Chapter 4 The Potential Equation 18.For the potential problem on an annular ring 1 r ∂ ∂r ( r∂ (^2) u ∂r ) +r^12 ∂ (^2) u ∂θ ...

Miscellaneous Exercises 289 24.Find a polynomial of second degree inxandy, v(x,y)=A+Bx+Cy+Dx^2 +Exy+Fy^2 , that satisfies the po ...

290 Chapter 4 The Potential Equation on the assumption thatis small anduis much smaller thanU 0. Using this boundary condition ...

Miscellaneous Exercises 291 b.Find the gradient ofuand plot some vectorsV=−grad(u)near the origin. u(x,y)=−tan−^1 (y x ) . Thefl ...

292 Chapter 4 The Potential Equation Figure 4 Exercise 38. of the lower strip (see Fig. 4). Of course,Vis bounded asy→∞.Solve th ...

Miscellaneous Exercises 293 h convection coefficient (W/m^2 K) κ thermal conductivity of steel (W/mK) L length of cooling line T ...

294 Chapter 4 The Potential Equation The problem in terms of dimensionless variables and parameters is: γ ∂θ ∂X= ∂^2 θ ∂X^2 + ∂^ ...

Higher Dimensions and Other Coordinates CHAPTER 5 5.1 Two-Dimensional Wave Equation: Derivation For an example of a two-dimensio ...

296 Chapter 5 Higher Dimensions and Other Coordinates (a) (b) Figure 2 (a) Distributed forces. (b) Concentrated forces. Figure 3 ...

Chapter 5 Higher Dimensions and Other Coordinates 297 Adding up forces in the vertical direction and equating the sum to the mas ...

298 Chapter 5 Higher Dimensions and Other Coordinates 2.Suppose that the frame is circular and that its equation isx^2 +y^2 =a^2 ...

5.2 Three-Dimensional Heat Equation 299 Figure 6 The heat flow rate through a small section of surface with area Ais q·nˆ A. sim ...

300 Chapter 5 Higher Dimensions and Other Coordinates Because the subregionVwas arbitrary, we conclude that the integrand must b ...

5.2 Three-Dimensional Heat Equation 301 Again using Fourier’s law, we obtain the boundary condition κ∂∂un(P,t)+hu(P,t)=hT(P,t) f ...

302 Chapter 5 Higher Dimensions and Other Coordinates Because differentiation with respect tox,y,ortgives the same result inside ...

5.3 Two-Dimensional Heat Equation: Solution 303 EXERCISES For the functionu(x,y,z,t)that satisfies Eqs. (10)–(14), show that ∫c ...

304 Chapter 5 Higher Dimensions and Other Coordinates variables by seeking solutions in the form u(x,y,t)=φ(x,y)T(t). On substit ...

5.3 Two-Dimensional Heat Equation: Solution 305 The sum of a function ofxand a function ofycan be constant only if those two fun ...

306 Chapter 5 Higher Dimensions and Other Coordinates and the corresponding functionTis Tmn=exp ( −λ^2 mnkt ) . We now begin to ...

5.3 Two-Dimensional Heat Equation: Solution 307 Thecoefficientsareeasilyfoundtobe amn= 4 ab π^2 cos(mπ)cos(nπ) mn = 4 ab π^2 (− ...