George Biddell Airy, K.C.B., M.A., LL.D., D.C.L.'s An elementary treatise on partial differential equations. PDF

By George Biddell Airy, K.C.B., M.A., LL.D., D.C.L.

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Additional resources for An elementary treatise on partial differential equations. Designed for the use of students in the university (2nd edition, 1873)

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17a) = f (ξ)e−λσ . 17b) = σ. 14): 1= 1 f (ξ) [f (ξ)e−λtc + a − (f (ξ) + a)] λ f (ξ) or λ = f (ξ)(e−λtc − 1). The earliest time for shock formation, therefore, is tc = 1 f (ξ) ln λ f (ξ) + λ . 18) min For tc to be positive we must have −f (ξ) > λ for some 0 ≤ ξ < X. 17a) for all t ≥ 0. It decays exponentially as t → ∞. 17b) gives σ x = ξ+ u(x(τ ), τ )dτ 0 σ f (ξ)e−λτ dτ = ξ+ 0 = ξ+ f (ξ) (1 − e−λσ ). λ Recalling that σ = t, we have x→ξ+ ©2000 CRC Press LLC f (ξ) as t → ∞. λ From this it follows that in the limit t → ∞, x ≤ xm = ξ + 1 f (ξ) λ , 0 ≤ ξ ≤ X.

21) yielding t0 = − 1 (α + 1)λX ln 1 − . 22) Thus, the rarefaction wave catches up with the shock only if 1− (α + 1)λX > 0, hα that is, 1 h > [(α + 1)λX] α . 22). Suppose a characteristic in the rarefaction has a value u = c at time t. 17), we have dx = (ce−λt )α = cα e−λαt dt which, on integration and use of the IC x = 0, t = 0, gives cα (1 − e−λαt ). λα The solution in the rarefaction wave is x= u(x, t) = ce−λt = λαx eλαt − 1 1 α . 24) After t = t0 , the motion of the shock is given by dxs dt = = uα λαxs b = α+1 (α + 1)(eλαt − 1) λαe−λαt xs .

67), and integrate with respect to x to get B (t) α(x, t) = A(t) − xt + B(t)Ω3 (η). 69) and have α(x, t) = A(t) − xt B (t) . 70) Since Γ3 (η) = Ω3 (η), we also have Γ3 (η) ≡ 0. 61), we get t + A(t) − xt B (t) − B(t)Γ1 (η) ηx + tηt = 0. 72) are dt dη dx = = . 73) is clearly η = constant; this is the similarity variable. 73), we have dx B (t) A(t) − lB(t) + x=1+ . 75) gives the similarity variable η = xB(t) − ©2000 CRC Press LLC 1+ A(t) − lB(t) B(t)dt . 70), we get − B (t) tB (t) B(t) B(t) B (t) B (t) B 2 (t) +t A (t) − x − xt + xt 2 B(t) B(t) B (t) t + A(t) − xt −x + t = B 3 (t)Γ4 (η).

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An elementary treatise on partial differential equations. Designed for the use of students in the university (2nd edition, 1873) by George Biddell Airy, K.C.B., M.A., LL.D., D.C.L.


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