By David M. Young, Robert Todd Gregory, Mathematics
Evaluation of user-friendly functions
Solution of a unmarried nonlinear equation with particular connection with polynomial equations
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Ideal as a path textual content in numerical research or as a supplementary textual content in numerical equipment, A Survey of Numerical Mathematics judiciously blends arithmetic, numerical research, and computation. the result's an surprisingly useful reference and studying device for contemporary mathematicians, computing device scientists, programmers, engineers, and actual scientists.
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Extra resources for A Survey of Numerical Mathematics [Vol I]
1 d du ρ2 2 ρ dρ dρ u(c) = 0, = f (ρ), 0 ≤ ρ < c, u(ρ) bounded at ρ = 0. 2 6. 7. 1 d u 1 du − 2 u = f (x), 0 ≤ x < a, + 2 dx x dx 4x u(a) = 0, u(x) bounded at x = 0. d2 u − γ 2 u = f (x), 0 < x, dx2 u(0) = 0, u(x) bounded as x → ∞. 50 8. Chapter 0 Ordinary Differential Equations d2 u − γ 2 u = f (x), −∞ < x < ∞, dx2 u(x) bounded as x → ±∞. 9. Use the Green’s function of Exercise 5 to solve the problem du 1 d ρ2 2 ρ dρ dρ = 1, 0 ≤ ρ < c, u(c) = 0, and compare with the solution found by integrating the equation directly.
Then t up (t) = G(t, z)f (z) dz t0 is a particular solution of the nonhomogeneous equation du d2 u + k(t) + p(t)u = f (t), dt 2 dt (NH) where G is the Green’s function defined by G(t, z) = u1 (z)u2 (t) − u2 (z)u1 (t) . W(z) EXERCISES In Exercises 1–10, find the general solution of the differential equation. (23) 24 Chapter 0 Ordinary Differential Equations du + a(u − T) = 0. dt du + au = e−at . 3. dt d2 u 5. + u = cos(t). dt 2 du + au = eat . dt d2 u 4. + u = cos(ωt) (ω = 1). dt 2 d2 u 6. − γ 2 (u − U) = 0 dx2 (U, γ 2 are constants).
4. (Critical radius) The neutron flux u in a sphere of uranium obeys the dif- ferential equation λ 1 d du ρ2 2 3 ρ dρ dρ + (k − 1) Au = 0 in the range 0 < ρ < a, where λ is the effective distance traveled by a neutron between collisions, A is called the absorption cross section, and k is the number of neutrons produced by a collision during fission. In addition, the neutron flux at the boundary of the sphere is 0. Make the substitution u = v/ρ and 3(k − 1)A/λ = µ2 , and determine the differential equation satisfied by v(ρ).
A Survey of Numerical Mathematics [Vol I] by David M. Young, Robert Todd Gregory, Mathematics