By A.V.; Kalinichenko, D.F. Bitsadze
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Additional info for A Collection of Problems on the Equations of Mathematical Physics
15) which gives the solution for the parameter α α = αn = nπ , n = 1, 2, 3, . . 16) Note that n = 0 (α = 0) leads to the trivial solution u (x, t) = 0, and hence, the case n = 0 is to be excluded. 16) that there exists an infinite set of discrete values of α for which the problem has a nontrivial solution. These values αn are called the eigenvalues, and the corresponding solutions are Xn (x) = Bn sin nπx . 18) where Cn and Dn are constants of integration. 19) where an = Cn Bn , bn = Bn Dn are new arbitrary constants and n = 1, 2, 3, .
25) are called the nth normal modes of vibration or the nth harmonic, and ωn represents the discrete spectrum of circular (or radian) frequency or νn = ω2πn = nc 2 , which are called the angular frequencies. The first harmonic (n = 1) is called the fundamental harmonic and all other harmonics (n > 1) are called overtones. The frequency of the fundamental mode is given by ω1 = πc , ν1 = 1 2 T∗ . 26ab) is considered the fundamental law (or Mersenne law) of a stringed musical instrument. 4 Several modes of vibration in a string.
Case III. B 2 − 4AC < 0. Equations for which B 2 − 4AC < 0 are called elliptic. 10ab) have no real solutions. So there are two families of complex characteristics. 17ab) so that ξ = α + iβ and η = α − iβ. 3). It can easily be verified that A∗ = 0 and C ∗ = 0 take the form A∗∗ − C ∗∗ +iB ∗∗ = 0 which are satisfied if and only if A∗∗ = C ∗∗ and B ∗∗ = 0. 18) reduces to the form uαα + uββ = H4 = H5 (α, β, u, uα , uβ ) . 19) This is called the canonical form of the elliptic equation. 1) is called hyperbolic, parabolic, or elliptic at a point (x0 , y0 ) accordingly as B 2 (x0 , y0 ) − 4A (x0 , y0 ) C (x0 , y0 ) > = < 0.
A Collection of Problems on the Equations of Mathematical Physics by A.V.; Kalinichenko, D.F. Bitsadze