tailieunhanh - Handbook of mathematics for engineers and scienteists part 139

Handbook of mathematics for engineers and scienteists part 139. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 934 Difference Equations and Other Functional Equations where B1 . B4 are arbitrary constants and k1 and k2 are the roots of the quadratic equation k - Ai k - A4 - A2A3 0. In the degenerate case ki k2 the terms ek- and e k2z in should be replaced with ipeki and ze k1 z respectively. In the case of purely imaginary or complex roots one should separate the real or imaginary part of the roots in solution . On substituting into the original functional equation one obtains conditions for the free coefficients and identifies the function f t namely B2 B4 0 f t A2A3 ki-Ai 2 Bii .- Bi B3 0 f t A2A3 k2-Ai 2 B2IA k2 Ai 0 f t A2A3 k2 BiB3e ki A2A3 k2BBe Solution involves arbitrary functions y x and t . Degenerate case. In addition the functional equation has two degenerate solutions formulas are used f BiB2eAi g A2Bi e Ai h BeAi R lh A z - A2Q where p Q x t and Q Q z are arbitrary functions Ai A2 Bi and B2 are arbitrary constants and f BiB2eAi h -Bie Ai - A2g Q A2B2eAiZ R B2eAiZ where p Q x t and g g x are arbitrary functions and Ai A2 Bi and B2 are arbitrary constants. 3 . Consider a more general functional equation of the form f t gi x Qi z gn x Qn z 0 where z Q x t . By differentiation in x this equation can be reduced to a functional differential equation which may be regarded as a bilinear functional equation of the form . Using formulas for the construction of its solution one can first obtain a system of ODEs and then find solutions of the original equation . 4 . Consider a functional equation of the form fi t gi x fm t gm x hi x Qi z hn x Qn z 0 z Q x t . Assume that gm x 0. Dividing equation by gm x and differentiating the result in x we come to an equation of the form 2n fi t gi x fm-i t gm-i x si x Ri z 0 with a smaller number of functions fi t . Proceeding in this way we can eliminate all functions fi t and obtain a .