Lectures on Differential Equations of Mathematical Physics: by Gerhard Freiling

By Gerhard Freiling

The speculation of partial differential equations of mathematical physics has been probably the most vital fields of research in utilized arithmetic. this is often primarily as a result widespread incidence of partial differential equations in lots of branches of average sciences and engineering. the current lecture notes were written for the aim of offering an method established almost always at the mathematical difficulties and their comparable recommendations. the first difficulty, for that reason, isn't with the overall thought, yet to supply scholars with the elemental options, the underlying ideas, and the recommendations and strategies of answer of partial differential equations of mathematical physics. one of many authors major targets is to give a reasonably effortless and whole advent to this topic that is compatible for the 'first analyzing' and available for college students of other specialities. the fabric in those lecture notes has been built and prolonged from a collection of lectures given at Saratov kingdom collage and displays in part the study pursuits of the authors. it truly is meant for graduate and complex undergraduate scholars in utilized arithmetic, computing device sciences, physics, engineering, and different specialities. the must haves for its examine are a customary easy path in mathematical research or complex calculus, together with basic usual differential equations. even though a variety of differential equations and difficulties thought of in those lecture notes are bodily encouraged, an information of the physics concerned isn't really priceless for figuring out the mathematical elements of the answer of those difficulties.

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Without loss of generality one can assume that a = b = 0 (this can always be achieved by the replacement u(x,t) = u(x,t) ˜ exp(−ax/2+bt/2) ). Thus, we consider the following Cauchy problem uxx − utt + cu = f (x,t), −∞ < x < ∞, t > 0; u|t=0 = ϕ(x), c = 0, ut|t=0 = ψ(x). 13) For the Riemann function we have the Goursat problem vxx − vtt + cv = 0, v|t=x−x0 +t0 = 1, v|t=−x+x0 +t0 = 1. 14) in the form v(x,t) = w(z), where z = (t − t0 )2 − (x − x0 )2 . Clearly, if the point (x,t) lies on the characteristics I1 or I2 , then z = 0, moreover z > 0 inside the triangular ∆(x0 ,t0 ).

Moreover, ux (x, y) = ψ (x) + = ψ (x) + Z y y0 Z y y0 wx (x, η) dη (av + bw + cu + f )(x, η) dη = v(x, y). Hyperbolic Partial Differential Equations 47 Thus ux = v. 1). e. 2). 4). 4) by the method of successive approximations. Put v0 (x, y) = ψ (x) + Z y w0 (x, y) = ϕ (y) + y0 Z x x0 f (x, η) dη, f (ξ, y) dξ, u0 (x, y) = ψ(x), Z y   (avn + bwn + cun )(x, η) dη,    y0   Z x  wn+1 (x, y) = (avn + bwn + cun )(ξ, y) dξ,  x0   Z y     un+1 (x, y) = wn (x, η) dη. 5) y0 Take the constants M ≥ 0 and K ≥ 1 such that |u0 |, |v0 |, |w0 | ≤ M, |a| + |b| + |c| ≤ K.

22) u|t=0 = ut|t=0 = 0. We consider the integral 1 2 E(t) = Z l 0 1 (ut (x,t))2 + (ux (x,t))2 a2 dx, which is called the energy integral. We calculate ˙ = E(t) ˙ := where E(t) d dt E(t). ˙ = E(t) Z l 0 Z l 1 ut (x,t)utt (x,t) + ux (x,t)uxt (x,t) dx, a2 0 Integrating by parts the second term in the last integral we obtain 1 utt (x,t) − uxx (x,t) dx + l0 (ux (x,t)ut (x,t)) . 22), and consequently, 1 utt (x,t) − uxx (x,t) ≡ 0, a2 ˙ = (ux ut )|x=l − (ux ut )|x=0 . 22) u(0,t) ≡ 0 and u(l,t) ≡ 0. Hence ut|x=0 = 0 and ut|x=l = 0.

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