# Download e-book for iPad: An Introduction to the Study of Integral Equations by M. Bocher By M. Bocher

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Extra info for An Introduction to the Study of Integral Equations

Example text

3 au — sinO— =0. sinO ao ao PROBLEMS u(x) denote the steady-state temperatures in a slab bounded by the planes x = 0 and x = c when those faces are kept at fixed temperatures u = 0 and u = u0, 1. Let respectively. Set up the boundary value problem for u(x) and solve it to show that and where is the flux of heat to the left across each plane x = x0 (0 x0 c c). 14 PARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS CHAP. 1 2. A slab occupies the region 0 c x c c. There is a constant flux of heat into the slab through the face x = 0.

9) that when A, B, . , F are constants and G 0, equation (1) always has solutions of the form U = exp (Ax + ,ay), where the constants A and ,a satisfy the algebraic . equation (2) From analytic geometry, we know that such an equation represents a conic plane and that the different types of conic sections arising are section in the similarly determined by B2 — 4AC. EXAMPLES. Laplace's equation uxx+uyy=0 is a special case of equation (1) in which A = C = 1 and B = elliptic throughout the xy plane.

16). To be specific, the quotient hf — g112/(b — a) is the mean, or average, value of the squares of the vertical distances If(x) — g(x)I between points on those graphs over the interval a < x < b. The quantity hf — gM2 is called the mean square deviation of one of the functions f and g from the other. y 0 b a Two functions f and g in x FIGURE16 b) are orthogonal when (f,g) =0, or (10) ff(x)g(x)dx=0. 44 FOURIER SERIES CHAP 2 the function f is said to be normalized. We have carried our analogy too far to preserve the original meaning of the geometric terminology.