(i) Solutions for higher order difference equations:High order difference homogeneous equation: 1 can still be the solution of the homogeneous equation2 get the corresponding characteristic equation (in fact, we can write the corresponding characteristic equation directly, and refer to the characteristic equation of the high number of the differential equation)there will be n characteristic roots (different real roots, multiple roots, conjugate complex roots)(1) dissimilar real roots :(2) real roots, M heavy root :Here is just one example, too complex to not elaborate, that is: Alpha 1 to alpha M are equal, but Alpha m+1 to Alpha N are different real roots. (In fact, it is not enough generalization) (note that the last one because of equality so Alpha 1 rewrite for Alpha M) (and he did not write the remainder of the real roots part, easy to cause misunderstanding) so the above statement is ambiguous. Because if there are 5 of the root, Alpha 1,2,3,4,5. But their weight may be different, such as 2,2,3,4,5. So the above statement is problematic. should read:here the n is replaced by T, here is only assuming that Alpha 1 is K-weight, while the rest of the k+1 to n-roots are different real roots. then the rest of how to write, you know. See two examples:(3) complex rootThe root of the complex root of the situation is no longer given(ii) Stability conditionsto get the above conclusion, we need to prove the following formula:then the above conclusions can be deduced from this formula. (I have proved and deduced in the detailed notes)(iii) non-Ziezet solutionthe form of the nonhomogeneous difference equation and the push process X (t)that the driving process is a deterministic processThe following are some examples of situations where the push process contains a constant term, a time trend item T. (1) (High-order difference equation)the equation at this point is:The form of the conjecture solution is:substituting equation, the value of the solution C is:but the denominator may be 0, then C will not exist. at this point, we should guess the form of the solution is:substituting Equation , the value of the solution C is: if the denominator is likely to be 0, then C will not exist. continue to try This form of solution, know to find so far, always find. (2) where b,d,r are constants (only one order is indicated) the equation at this point is: we only consider the first order:the form of conjecture solution is:substituting equation, contact c0,c1 So, get a special solution for:as long as the |d^r|<1 , the solution converges namely:1 (constant term multiplied by T)2, try usingeach item is multiplied by T for higher order equations, this method can still be used (although the first solution of conjecture requires some wisdom) (3) where B is a constant and D is the normal number. the equation at this point is: It is assumed that the general form of its special solution is:give an example of a second-order difference equation:the Equation form is: (At this time d=1) It is assumed that the general form of its special solution is: the surrogate can to two coefficients for:Similarly, the special solution form at this time is:(iv) Undetermined coefficient method(the undetermined coefficient method is also commonly used in differential equations, first guessing a challenge solution, assuming that it satisfies, then substituting, finally to find these coefficients, if the coefficients have a solution, then this challenge solution is the solution of the equation, if the coefficient is not solved, then the solution of the challenge is not the solution of the equation ) that is, the driving process is a special solution of the non-homogeneous equation of stochastic disturbance term.The undetermined coefficients method may be misunderstood, so we will start with the solution of the trial as a challenge solution (1) Simple Case 1: First Order difference equation + a stochastic disturbance term equation is The Challenge solution form of conjecture is: The surrogate equation has to be for any T and Efshow, the above formula must be set up, it can only let the constant term and coefficients are zero ~ so you can get:and theconsider the denominator, or the classification discussion:Classification Scenario 1: The result is exactly the same as the first part using the forward iterative Jiuben equation for the resulting solution.Finally, we can match the general solution of the corresponding homogeneous equation and combine it into the general solution of the non-homogeneous equation, as follows: Classification Scenario 2: since the summation of efshow is not necessarily limited, the solution may diverge. Therefore, the following conditions are applied: Finally, the special solution is written as: but I think, because of the existence of T, the solution is still divergent, so the initial conditions are applied and the egg. (2) Simple case 2: First order difference equation + two random disturbance termthe equation is: The Challenge solution form of conjecture is: use the steps and Methods in (1) Simple Case 1 and don't repeat them. (3) Second order difference equation + a random disturbance term, the equation is: The Challenge solution form of conjecture is: The surrogate equation can be obtained: and theand (Can be solved A (j)) (v) Lag operatorhysteresis operator L: Properties of hysteresis operators: The difference equation can be solved by using properties 5 and properties 6, combined with property 1. (Progression summation and expansion)if it is higher, then it can be factorization, split, then the series expansion.
from for Notes (Wiz)
Chapter I time series Foundation--Difference equation and solution (II.)