tailieunhanh - Handbook of mathematics for engineers and scienteists part 93

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 93', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 612 Linear Partial Differential Equations Given the transform f p the function f t can be found by means of the inverse Laplace transform 1 f c iœ f t L-1 p where L-1 p s f p ept dp 2nï Jc-iœ where the integration path is parallel to the imaginary axis and lies to the right of all singularities of f p which corresponds to C CTq. In order to solve nonstationary boundary value problems the following Laplace transform formulas for derivatives will be required L f t P P - f 0 L f t p2 p - pf 0 - f 0 where f 0 and f 0 are the initial conditions. More details on the properties of the Laplace transform and the inverse Laplace transform can be found in Section . The Laplace transforms of some functions are listed in Section . Tables of inverse Laplace transforms are listed in Section . Such tables are convenient to use in solving linear problems for partial differential equations. . Solution procedure for linear problems using the Laplace transform. Figure shows schematically how one can utilize the Laplace transforms to solve boundary value problems for linear parabolic or hyperbolic equations with two independent variables in the case where the equation coefficients are independent of t the same procedure can be applied for solving linear problems characterized by higher-order equations . Here and henceforth the short notation w x p L w x t will be used the arguments of w may be omitted. It is significant that with the Laplace transform the original problem for a partial differential equation is reduced to a simpler problem for an ordinary differential equation with parameter p the derivatives with respect to t are replaced by appropriate algebraic expressions taking into account the initial conditions see formulas . . Solving linear problems for parabolic equations with the Laplace transform. Consider a linear nonstationary boundary value problem for the parabolic equation dw dt a x -W b x W c x w x t dx2 dx .