By George Biddell Airy, K.C.B., M.A., LL.D., D.C.L.
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This booklet provides chosen issues in technology and engineering from an applied-mathematics perspective. The defined average, socioeconomic, and engineering phenomena are modeled by way of partial differential equations that relate country variables equivalent to mass, speed, and effort to their spatial and temporal adaptations.
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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)
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).
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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