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Handbook of mathematics for engineers and scienteists part 95

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 95', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 626 Linear Partial Differential Equations 14.8.3. Problems for Equation dF at 9w_ b t _ v x 9wJ dt 0 t dx p x dx - g x w x t 14.8.3-1. General relations to solve nonhomogeneous boundary value problems. Consider the generalized telegraph equation of the form T a t G b t T p x TT 1 _ x w x t . dt2 dt t dx v dx J J 14.8.3.1 It is assumed that the functions p p x and q are continuous and p 0 for x1 x x2. The solution of equation 14.8.3.1 under the general initial conditions 14.8.1.2 and the arbitrary linear nonhomogeneous boundary conditions 14.8.1.3 - 14.8.1.4 can be represented as the sum nX2 r G x t t r dt dr 1 x2 X2 dt fi t a O fo t G x t 1 0 dt Jxi t d drG x t t t X pfa g1 r b r A1 x t r dr p x2 g2 r b r A2 x t r dr . 14.8.3.2 Jo Jo Here the modified Green s function is determined by - . yn x yn t i i x G x t t T ---ü--Ü2---Un t T n 1 IUn 1 fX I Un x dx 14.8.3.3 Xi where the An and yn x are the eigenvalues and corresponding eigenfunctions of the Sturm-Liouville problem for the following second-order linear ordinary differential equation with homogeneous boundary conditions p x yX X A - q x y 0 a1yX fay 0 at x x1 14.8.3.4 a2yX fay 0 at x x2. The functions Un Un t r are determined by solving the Cauchy problem for the linear ordinary differential equation Un a t Un Anb t Un 0 i . i 14.8.3.5 U .r 0 Uni_ i. The prime denotes the derivative with respect to t and r is a free parameter occurring in the initial conditions. The functions A1 x t and A2 x t that occur in the integrands of the last two terms in solution 14.8.3.2 are expressed in terms of the Green s function of 14.8.3.3 . The corresponding formulas will be specified below when studying specific boundary value problems. The general and special properties of the Sturm-Liouville problem 14.8.3.4 are detailed in Subsection 12.2.5. Asymptotic and approximate formulas for eigenvalues and eigenfunctions are also presented there. 14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable 627 .