New PDF release: A Survey of Numerical Mathematics [Vol I]

By David M. Young, Robert Todd Gregory, Mathematics

Designed as a textual content for a three-semester direction in numerical research, this extremely popular paintings provides a vast review of computer-oriented numerical algorithms for fixing a variety of sorts of mathematical difficulties. The e-book is meant not just to impart a operating wisdom of sensible strategies for fixing genuine difficulties yet to organize the coed for complex reviews on the graduate level.
Topics include:
Evaluation of user-friendly functions
Solution of a unmarried nonlinear equation with particular connection with polynomial equations
Interpolation and approximation
Numerical differentiation and quadrature
Ordinary differential equations
Computational difficulties in linear algebra
Numerical resolution of elliptic and parabolic partial differential equations through finite distinction methods
Solution of enormous linear structures by way of iterative methods
In addition to thorough insurance of the basics, those wide-ranging volumes include such detailed positive aspects as an advent to computing device mathematics, together with an errors research of a process of linear algebraic equations with rational coefficients, and an emphasis on computations in addition to mathematical elements of varied problems.
Geared towards senior-level undergraduates and first-year graduate scholars, the ebook assumes a few wisdom of complex calculus, common complicated research, matrix concept, and traditional and partial differential equations. even if, the paintings is basically self-contained, with easy fabric summarized in an appendix, making it an ideal source for self-study.
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]

Sample text

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).

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A Survey of Numerical Mathematics [Vol I] by David M. Young, Robert Todd Gregory, Mathematics


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