## Download Applied partial differential equations. An introduction by Alan Jeffrey PDF

By Alan Jeffrey

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**Additional resources for Applied partial differential equations. An introduction**

**Example text**

8 shows several wave proﬁles that indicate steepening of the signal as it propogates. At t = 1 the wave breaks, which is the ﬁrst instant when the solution would become multiple valued. To ﬁnd the solution for t < 1 we note that u(x, t) = 2 for x < 2t and u(x, t) = 1 for x > t + 1. 19) becomes x = (2 − ξ)t + ξ, which gives ξ= x − 2t . 20) then yields u(x, t) = 2−x , 1−t 2t < x < t + 1, t < 1. This explicit form of the solution also indicates the diﬃculty at the breaking time t = 1. 17) may have a solution only up to a ﬁnite time tb , which is called the breaking time.

13. Write a formula for the general solution of the equation ut + cux = f (x)u. Hint: Your answer should involve an integral with variable limits of integration. 14. Consider the Cauchy problem ut = xuux , x ∈ R, t > 0, x ∈ R. u(x, 0) = x, Find the characteristics, and ﬁnd a formula that determines the solution u = u(x, t) implicitly as a function of x and t. Does a smooth solution exist for all t > 0? 15. Consider the initial value problem x ∈ R, t > 0, ut + uux = 0, u(x, 0) = 1 − x2 , |x| ≤ 1, 0, |x| > 1.

Find the steady-state concentration and sketch its spatial proﬁle for diﬀerent values of L and λ. Hint: Break the problem up into two parts, on each side of the source. At the source the concentration must be continuous, and in a small interval about the source, the ‘ﬂux in’ minus the ‘ﬂux out’ equals one. 5. Work the preceding problem if there is no decay and if the source is located at x = ξ, 0 < ξ < L. 6. Consider the advection-diﬀusion equation on an interval 0 < x < L. Show that if the ﬂux at x = 0 equals the ﬂux at x = L, then the density is constant.