Partial Differential Equations |
Contents
The Electric Circuit | 4 |
Remarks on the MultiDimensional Case | 5 |
The Problem of Cauchy for Hyperbolic Equations | 12 |
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५६६ absolutely integrable adjoint arbitrary Bessel's inequality boundary conditions C₁ Cauchy conditions Cauchy problem characteristic values coefficients conditions u(x,0 consider constant continuous corresponding curve define denumerable differential equation discontinuities displacement domain exists expression follows formula Fourier series Fourier transform function f fundamental functions given gives Green function Hence homogenous equation hyperbolic identity independent initial conditions interval L₁ L₁(y L₂ Laplace Laplace transform left member linear Newton's second law non-homogenous obtained orthogonal partial derivatives region respect Riemann function right member satisfies second order self-adjoint solution of 3.10 solve step functions string Substituting suppose t-to tangential derivative theorem trigonometric series u₁ uniformly convergent vanish variables velocity vibrations VL(u whence y₁ yields Young's modulus zero αλ αξ ίλ бут