By Gerhard Freiling
The idea of partial differential equations of mathematical physics has been the most very important fields of analysis in utilized arithmetic. this can be primarily a result of common incidence of partial differential equations in lots of branches of usual sciences and engineering. the current lecture notes were written for the aim of featuring an technique dependent commonly at the mathematical difficulties and their similar options. the first main issue, for this reason, isn't with the final thought, yet to supply scholars with the basic options, the underlying ideas, and the suggestions and techniques of resolution of partial differential equations of mathematical physics. one of many authors major targets is to provide a pretty simple and entire creation to this topic that is compatible for the 'first analyzing' and obtainable for college kids of alternative specialities. the fabric in those lecture notes has been built and prolonged from a collection of lectures given at Saratov country collage and displays partly the examine pursuits of the authors. it really is meant for graduate and complex undergraduate scholars in utilized arithmetic, laptop sciences, physics, engineering, and different specialities. the must haves for its learn are a typical simple direction in mathematical research or complicated calculus, together with basic usual differential equations. even though numerous differential equations and difficulties thought of in those lecture notes are bodily prompted, a data of the physics concerned isn't invaluable for knowing the mathematical elements of the answer of those difficulties.
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Freiling and V. Yurko 28 3. 23) u|t=0 = ϕ(x), ut|t=0 = ψ(x). Let the function f (x,t) be continuous in D, twice continuously differentiable with respect to x, and f (0,t) = f (l,t) = 0. 1. We expand the function f (x,t) into a Fourier series with respect to x : ∞ f (x,t) = ∑ fn (t) sin n=1 where nπ x, l Z nπ 2 l f (x,t) sin x dx. 9): ∞ u(x,t) = ∑ un (t) sin n=1 nπ x. 18). 25) has a unique solution which can be found, for example, by the method of variations of constants: un (t) = An sin + l anπ Z t 0 anπ anπ t + Bn cos t l l fn (τ) sin anπ (t − τ) dτ.
Integrating by parts the first term we get Z I1 − Z P M = β1 (x)α1 (x) P M α1 (x) 2β1 (x) − (a − b)(x, x − x0 + t0 )β1 (x) dx. Impose a first condition on the function v(x,t), namely: 2β1 (x) − (a − b)(x, x − x0 + t0 )β1 (x) = 0. Solving this ordinary differential equation we calculate 1 v(x, x − x0 + t0 ) = exp 2 Then Z x x0 (a − b)(ξ, ξ − x0 + t0 ) dξ . 4) Z I1 = u(P)v(P) − u(M)v(M). 5) G. Freiling and V. Yurko 52 2) On I2 : t = −x + x0 + t0 , dt = −dx. Denote α2 (x) = u(x, −x + x0 + t0 ), β2 (x) = v(x, −x + x0 + t0 ).
9). e. 35) for the same value λ = λ0 . 2. 35). For definiteness, let below h1 H1 = 0. The other cases are considered analogously and can be recommended as exercises. 36). 36) without loss of generality. 38) where q(x) ∈ L2 (0, π), h and H are real. Denote y(x) := −y (x) + q(x)y(x), U(y) := y (0) − hy(0), V (y) := y (π) + Hy(π). 37) under the initial conditions ϕ(0, λ) = 1, ϕ (0, λ) = h, ψ(π, λ) = 1, ψ (π, λ) = −H. Hyperbolic Partial Differential Equations 31 For each fixed x, the functions ϕ(x, λ) and ψ(x, λ) are entire in λ.