By M. Bocher
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Extra info for An Introduction to the Study of Integral Equations
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.
An Introduction to the Study of Integral Equations by M. Bocher