solution of linear nth order odes with constant coefficients

... third -order nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol 294, no 1, pp 104–112, 2004 Y Feng, Solution and positive solution of a semilinear third -order equation,” ... third -order generalized right focal problem,” Journal of Mathematical Analysis and Applications, vol 288, no 1, pp 1–14, 2003 Z Du, W Ge, and X Lin, “Existence of solutions for a class of third -order ... monotone positive solution of the BVP 1.3 Theorem 3.2 If H1 f ∞ < < H2 f0 , then the BVP 1.3 has at least one monotone positive solution Proof The proof is similar to that of Theorem 3.1 and...

Ngày tải lên: 21/06/2014, 11:20

... Proof It follows from Theorem 2.1 and the left translation invariance of ᏸ The details are contained in [3, Proof of Theorem 3] From this theorem we obtain the proof of Theorem 3.1 Proof of Theorem ... one of the numerous extensions of the classical Gauss mean value theorem for harmonic functions For a proof of it, we directly refer to [6, Theorem 1.5] We would like to stress that in this proof ... open subset of Ωr (0,T) for every T,r > (see [2, Lemma 2.3]) This property of K(T, ·), together with the dλ -homogeneity of ᏸ, leads to the following Harnack-type inequality for entire solutions...

Ngày tải lên: 22/06/2014, 19:20

... Proof It follows from Theorem 2.1 and the left translation invariance of ᏸ The details are contained in [3, Proof of Theorem 3] From this theorem we obtain the proof of Theorem 3.1 Proof of Theorem ... one of the numerous extensions of the classical Gauss mean value theorem for harmonic functions For a proof of it, we directly refer to [6, Theorem 1.5] We would like to stress that in this proof ... open subset of Ωr (0,T) for every T,r > (see [2, Lemma 2.3]) This property of K(T, ·), together with the dλ -homogeneity of ᏸ, leads to the following Harnack-type inequality for entire solutions...

Ngày tải lên: 22/06/2014, 22:20

Ngày tải lên: 22/03/2014, 11:20

... is called a linear homogeneous functional equation of kth order with constant coefficients The coefficients are the constants a j , j = 0,1,2, ,k It is assumed that ak = A solution of (2.1) is a ... continuous solution of the associated Abel functional equation (2.2) Then the functions f = λX ∗ α(x) f = λX , ∗ α(x) , , fk = λX k ∗ α(x) , (2.4) are linearly independent solutions of (2.1) Proof Functions ... of the characteristic equation By the theorems above, it has a solution of the form f = λα(x) , (3.3) where α satisfies Abel equation (3.1) Linear system (3.2) is a linear difference equation with...

Ngày tải lên: 22/06/2014, 22:20

... Description of the problem considered The motivation of our investigation goes back to [10] dealing with the linear system of differential equations with constant coefficients and constant delay One of the ... this solution, in accordance with the theory of linear equations, as the sum of the solution of adjoint homogeneous problem (3.1), (3.2) (satisfying the same initial data) and a particular solution ... results of the paper With the aid of discrete matrix delayed exponential we give formulas for the solution of the homogeneous and nonhomogeneous problems (1.1), (1.2) 3.1 Representation of the solution...

Ngày tải lên: 22/06/2014, 22:20

... stability of the first -order linear recurrence in a Banach space Using some ideas from [7] in this paper, one obtains a result concerning the stability of the n -order linear recurrence with constant ... stability of a linear recurrence Proof Let ε > 0, and consider the sequence (xn )n≥0 , given by the recurrence xn+2 + xn+1 − 2xn = ε, n ≥ 0, x0 ,x1 ∈ K (2.30) n ≥ 0, (2.31) A particular solution of ... |a| n bk an−k−1 ≤ ε ≤ |a|k k=0 ε − |a| , (2.13) n ≥ The stability result for the p -order linear recurrence with constant coefficients is contained in the next theorem Theorem 2.3 Let X be a Banach...

Ngày tải lên: 23/06/2014, 00:20

... Introduction Much of the sophistication of complicated linear equation-solving packages” is devoted to the detection and/or correction of these two pathologies As you work with large linear sets of equations, ... can be either no solution, or else more than one solution vector x In the latter event, the solution space consists of a particular solution xp added to any linear combination of (typically) N ... , N by the reference a[i] Tasks of Computational Linear Algebra • Solution of the matrix equation A·x = b for an unknown vector x, where A is a square matrix of coefficients, raised dot denotes...

Ngày tải lên: 15/12/2013, 04:15

... two rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y Rather, it just corresponds to writing the same set of linear equations ... interchange corresponding rows of the x’s and of Y In other words, this interchange scrambles the order of the rows in the solution If we this, we will need to unscramble the solution by restoring the ... end of the main loop over columns of the reduction It only remains to unscramble the solution in view of the column interchanges We this by interchanging pairs of columns in the reverse order...

Ngày tải lên: 15/12/2013, 04:15

... simply the product of Q with the 2(N − 1) Jacobi rotations In applications we usually want QT , and the algorithm can easily be rearranged to work with this matrix instead of with Q Sample page ... purposes, because of its greater diagnostic capability in pathological cases.) Updating a QR decomposition Some numerical algorithms involve solving a succession of linear systems each of which differs ... solve linear systems In many applications only the part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv Sample page from NUMERICAL RECIPES IN C: THE ART OF...

Ngày tải lên: 15/12/2013, 04:15

... Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Suppose we are able to write the matrix A as a product of two matrices, L·U=A (2.3.1) ... equally small operations count, both for solution with any number of right-hand sides, and for matrix inversion For this reason we will not implement the method of Gaussian elimination as a routine ... and the increasing numbers of predictable zeros reduce the count to one-third), and N M times, respectively Each backsubstitution of a right-hand side is N executions of a similar loop (one multiplication...

Ngày tải lên: 24/12/2013, 12:16

... Computer Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations ... modify the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors ... backsubstitute with the columns of B instead of with the unit vectors that would give A’s inverse This saves a whole matrix multiplication, and is also more accurate The determinant of an LU decomposed...

Ngày tải lên: 24/12/2013, 12:16

... improved solution x 2.5 Iterative Improvement of a Solution to Linear Equations Obviously it is not easy to obtain greater precision for the solution of a linear set than the precision of your ... the solution of the linear system by LU decomposition can be accomplished much faster, and in much less storage, than for the general N × N case The precise definition of a band diagonal matrix with ... Unfortunately, for large sets of linear equations, it is not always easy to obtain precision equal to, or even comparable to, the computer’s limit In direct methods of solution, roundoff errors accumulate,...

Ngày tải lên: 24/12/2013, 12:16

... submatrices Imagine doing the inversion of a very large matrix, of order N = 2m , recursively by partitions in half At each step, halving the order doubles the number of inverse operations But this means ... “7/8”; it is that factor at each hierarchical level of the recursion In total it reduces the process of matrix multiplication to order N log2 instead of N What about all the extra additions in (2.11.3)–(2.11.4)? ... matrices The problem of multiplying two very large matrices (of order N = 2m for some integer m) can now be broken down recursively by partitioning the matrices into quarters, sixteenths, etc And note...

Ngày tải lên: 24/12/2013, 12:16

... inverse of the matrix A, so that B0 · A is approximately the identity matrix Define the residual matrix R of B0 as 58 Chapter Solution of Linear Algebraic Equations We can define the norm of a matrix ... discussion of the use of SVD in this application to Chapter 15, whose subject is the parametric modeling of data SVD methods are based on the following theorem of linear algebra, whose proof is beyond ... than the square root of your computer’s roundoff error, then after one application of equation (2.5.10) (that is, going from x0 ≡ B0 · b to x1 ) the first neglected term, of order R2 , will be smaller...

Ngày tải lên: 24/12/2013, 12:16

... Value Decomposition A A⋅x = b (a) null space of A solutions of A⋅x = d solutions of A ⋅ x = c′ SVD solution of A ⋅ x = c range of A d c′ c SVD solution of A⋅x = d (b) Figure 2.6.1 (a) A nonsingular ... same permutation of the columns of U, elements of W, and columns of V (or rows of VT ), or (ii) forming linear combinations of any columns of U and V whose corresponding elements of W happen to ... (order N instead of N ) and space (order N instead of N ) The method of solution was not different in principle from the general method of LU decomposition; it was just applied cleverly, and with...

Ngày tải lên: 24/12/2013, 12:16

... case of a tridiagonal matrix was treated specially, because that particular type of linear system admits a solution in only of order N operations, rather than of order N for the general linear ... applications.) • Each of the first N locations of ija stores the index of the array sa that contains the first off-diagonal element of the corresponding row of the matrix (If there are no off-diagonal elements ... are applicable to some general classes of sparse matrices, and which not necessarily depend on details of the pattern of sparsity 74 Chapter Solution of Linear Algebraic Equations (A + u ⊗ v)...

Ngày tải lên: 24/12/2013, 12:16

... says that Ajk is exactly the inverse of the matrix of components xk−1 , which i appears in (2.8.2), with the subscript as the column index Therefore the solution of (2.8.2) is just that matrix inverse ... and denominators of the specific Pj ’s via synthetic division by the one supernumerary term (See §5.3 for more on synthetic division.) Since each such division is only a process of order N , the ... actually two distinct sets of solutions to the original linear problem for a nonsymmetric matrix, namely right-hand solutions (which we have been discussing) and left-hand solutions zi The formalism...

Ngày tải lên: 21/01/2014, 18:20

... case of a tridiagonal matrix was treated specially, because that particular type of linear system admits a solution in only of order N operations, rather than of order N for the general linear ... says that Ajk is exactly the inverse of the matrix of components xk−1 , which i appears in (2.8.2), with the subscript as the column index Therefore the solution of (2.8.2) is just that matrix inverse ... and denominators of the specific Pj ’s via synthetic division by the one supernumerary term (See §5.3 for more on synthetic division.) Since each such division is only a process of order N , the...

Ngày tải lên: 21/01/2014, 18:20

... 126t − 30) = 4(t − 1)(20t3 − 30t2 + 78t + 15) < 4(t − 1)(−30t2 + 78t) < This completes the proof of the necessary condition We prove the sufficient condition Put a = − x and b = + x, where < x ... 19 ε(x) < −8x2 − x4 − x14 − x16 < 12 12 This completes the proof ε(x) = −8x2 − R EFERENCES [1] V CÎRTOAJE, On some inequalities with power-exponential functions, J Inequal Pure Appl Math., 10(1)...

Ngày tải lên: 15/08/2014, 16:55