Download e-book for iPad: A First Course in the Numerical Analysis of Differential by Arieh Iserles

By Arieh Iserles

ISBN-10: 0521556554

ISBN-13: 9780521556552

This booklet provides a rigorous account of the basics of numerical research of either traditional and partial differential equations. the purpose of departure is mathematical however the exposition strives to take care of a stability between theoretical, algorithmic and utilized features of the topic. intimately, themes coated contain numerical resolution of standard differential equations by way of multistep and Runge-Kutta tools; finite distinction and finite parts ideas for the Poisson equation; a number of algorithms to unravel huge, sparse algebraic platforms; and techniques for parabolic and hyperbolic differential equations and methods in their research. The booklet is observed by way of an appendix that provides short back-up in a couple of mathematical themes.

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Extra info for A First Course in the Numerical Analysis of Differential Equations

Example text

The contraction property (118) allows to show that the mapping Hε 1 maps W(r1 , R1 ) × BP1 X (0, r2 ) × BR (0, η ) into W(r1 , R1 ), while the facts that the range of P0 is finite-dimensional and is contained in Z allow to prove that Hε 2 maps W(r1 , R1 ) × BP1 X (0, r2 ) × BR (0, η ) into BP1 X (0, r2 ). In order to show that the mapping Hε (ϕ , ψ , τ ) admits a fixed point (ϕε , ψε , τε ) ∈ W(r1 , R1 ) × BP1 X (0, r2 ) × BR (0, η ), besides the hypotheses recalled earlier, one needs additional properties.

Since 0 ((Du F0 (ω0 , p0 ) − I)ϕε ) vanishes, we can write Kε (τ ) = d0 0 Fε (ω0 + τ , p0 + ϕε (τ )) − Fε (ω0 + τ , p0 ) − Du F0 (ω0 , p0 )ϕε (τ ) +Fε (ω0 + τ , p0 ) − F0(ω0 + τ , p0 ) +F0 (ω0 + τ , p0 ) − F0(ω0 , p0 ) − τ Dτ F0 (ω0 , p0 ) = K1 + K2 + K3 + K4 + K5 , (81) where K1 = d0 0 ( 1 0 Du Fε (ω0 + τ , p0 + sϕε ) − Du Fε (ω0 + τ , p0 ) ϕε (τ )ds), K2 = d0 0 ((Du Fε (ω0 + τ , p0 ) − Du F0 (ω0 + τ , p0 ))ϕε (τ )), K3 = d0 0 ((Du F0 (ω0 + τ , p0 ) − DuF0 (ω0 , p0 ))ϕε (τ )), K4 = d0 0 (Fε (ω0 + τ , p0 ) − F0(ω0 + τ , p0 )), K5 = d0 0 (F0 (ω0 + τ , p0 ) − F0(ω0 , p0 ) − τ Dτ F0 (ω0 , p0 )).

7. 8. Under the Hypotheses (H1)–(H5), for any r, 0 < r ≤ r0 , there exist positive constants ε1 (r) ≤ ε0 (r) and δ1 (r) ≤ δ0 (r) such that, for 0 < ε ≤ ε1 (r), the map Kε has a fixed point τε with |τε | ≤ δ1 (r), and therefore Fε (ω0 + τε , ·) has a fixed point p0 + ϕ (ε , ω0 + τε ) ∈ BCω (X) (p0 , r). 0 Moreover, there exists a positive constant C5 such that |τε | ≤ C5 ε β2 . K. Hale and G. Raugel Proof. Let 0 < r ≤ r0 be fixed. To show the existence of a fixed point of Kε , we shall apply the Leray fixed point theorem.

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A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles


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