solution methods of finite difference equations

... Accuracy of Finite Difference Solutions 3.9 Summary References Solution Methods of Finite Difference Equations 4.1 Elliptic Equations 4.1.1 Finite Difference Formulations 4.1.2 Iterative Solution Methods ... FINITE DIFFERENCE METHODS Derivation of Finite Difference Equations 3.1 Simple Methods 3.2 General Methods 3.3 Higher Order Derivatives 45 45 46 50 v vi CONTENTS 3.4 Multidimensional Finite Difference ... derivations of finite difference equations Simple methods are followed by general methods for higher order derivatives and other special cases Finite difference schemes and solution methods for...

Ngày tải lên: 01/04/2014, 10:45

... concepts of minimal solution and least solution (or maximal and greatest solutions), unfortunately often identified in the literature on lower and upper solutions First-order differential equations ... and compact set of fixed points in C(I), which are exactly the solutions of problem (4) Claim 2: Every solution x of (4) satisfies α ≤ x ≤ β on I and, therefore, it is a solution of (1) in [α, β] ... prove that all solutions x of (4) obey x ≤ β on I Claim 3: The set of solutions of problem (1) in [α, β] has maximal and minimal elements The set S = {x ∈ C(I) : x is a solution of (1) , x ∈ [α,...

Ngày tải lên: 20/06/2014, 20:20

... theorems on oscillation and nonoscillation of solutions to 1.3 are obtained Thus, solutions of two different equations and/or two different solutions of the same equation are compared, which allows ... Figure 1: The graph of 30 iterates for the solutions of 4.11 and 4.13 illustrates the result of Theorem 4.5, √ here y t > x t for all t ∈ 3, ∞ √N Corollary 4.8 Let x be a solution of 3.1 , and Y ... slowly oscillating solutions P2 In Section 4, oscillation properties of equations with different coefficients, delays and initial functions were compared, as well as two solutions of equations with...

Ngày tải lên: 21/06/2014, 11:20

... more precise description of the set of solutions of problem (PN ) Lemma 2.5 Assume conditions (H) and (H f ) Let u, v ∈ [α,β] be two solutions of problem (PN ) Then, one of the following statements ... following characterization of the set of solutions of problem (PN ) Corollary 3.2 Let u : JN → R, then u is a solution of problem (PN ) if and only if u is a fixed point of operator T Proposition ... set of zeros of f is the ternary Cantor set 342 Stability of periodic solutions If we look for the constant solutions of problem ∆u(n) = f u(n) , n ∈ N, (4.4) we know that they are the zeros of...

Ngày tải lên: 23/06/2014, 00:20

... Advances in Difference Equations 2011, 2011:14 http://www.advancesindifferenceequations.com/content/2011/1/14 Page of 17 It completes the proof Now consider the existence of solutions of PBVP (1.1) ... Difference Equations 2011, 2011:14 http://www.advancesindifferenceequations.com/content/2011/1/14 Page 11 of 17 2.5 1.5 0.5 −0.5 −1 0.5 1.5 2.5 Figure Figure of Example 2.3 has at least one solution ... x(t) is a solution of (3.6) By the product rule we have Xing and Fu Advances in Difference Equations 2011, 2011:14 http://www.advancesindifferenceequations.com/content/2011/1/14 Page 13 of 17 r...

Ngày tải lên: 21/06/2014, 02:20

... cone, in this article, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained ... types of intervals are defined similarly The theory of impulsive differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of ... boundary value problems of impulsive dynamic equations on time scales [22–36] However, to the best of our knowledge, few papers concerning PBVPs of impulsive dynamic equations on time scales...

Ngày tải lên: 21/06/2014, 17:20

... extremal solutions of 1.1 are given as the infimum of all upper solutions and the supremum of all lower solutions, that is, the least solution of 1.1 is given by inf u t : u upper solution of 1.1 ... Upper Solutions Another modification of the concepts of lower and upper solutions concerns the possibility of allowing jumps in their graphs Since the task of finding a pair of lower and upper solutions ... least solution of 1.1 in α, β is given by x∗ t inf u t : u upper solution of 1.1 , u ∈ α, β , t ∈ I, 3.3 t ∈ I 3.4 and the greatest solution of 1.1 in α, β is given by x∗ t sup l t : l lower solution...

Ngày tải lên: 21/06/2014, 20:20

... difference equations with impulses,” International Journal of Difference Equations, vol 1, no 2, pp 225–239, 2006 15 Y Li, X Fan, and L Zhao, “Positive periodic solutions of functional differential equations ... problems of first-order differential equations, ” Journal of Mathematical Analysis and Applications, vol 325, no 1, pp 226–236, 2007 Advances in Difference Equations 14 J Li and J Shen, “Positive solutions ... problems for impulsive differential equations and impulsive difference equations have received much attention 9–19 On the other hand, recently, the theory of dynamic equations on time scales has become...

Ngày tải lên: 21/06/2014, 20:20

... conditions, solutions of inequalities are greater or less than solution of the equation can be formulated in the following form of positivity of the Cauchy function C t, s and the solution X t ... t, ξ f t , t ∈ 0, ∞ 2.3 Journal of Inequalities and Applications We can study properties of solution of 2.3 on each finite interval 0, ω since every solution x t of 2.1 satisfies also the equation ... the solution y t of 2.11 with the initial condition y s and the solution x t of 2.33 coincide for t ∈ s, ω Positivity of the operator S, the inequalities ρ S < and 2.31 imply nonnegativity of...

Ngày tải lên: 22/06/2014, 03:20

... there exists n0 ∈ N such that 2.11 holds, then every solution of 1.9 oscillates on t0 , ∞ T Proof The proof is an immediate consequence of Lemmas 2.1 and 2.2 We need the following lemmas in ... 2.26 and 2.11 hold Then, every solution of 1.9 is oscillatory on t0 , ∞ T Proof The proof follows from Lemmas 2.1, 2.2, and 2.5 Remark 2.7 We obtain the main results of 7, by letting n0 in Theorem ... We state the following lemma, which is an extension of 3, Lemma and improvement of 10, Lemma Lemma 2.1 Let x be a nonoscillatory solution of 1.9 If t lim sup t→∞ p η Δη > 0, 2.1 τ t then lim...

Ngày tải lên: 22/06/2014, 11:20

... that the results of this paper are presented in a form which is essentially new and of high degree of generality Also, for related results in oscillation of forced differential equations, we refer ... by b(n), then (1.1) is oscillatory Ravi P Agarwal et al Proof Let {x(n)} be an eventually positive solution of (1.1) As in the proof of Theorem 2.2, we obtain (2.8) Now m−1 H(m,n)e(n) H(m,k) ... n→∞ n→∞ then all bounded solutions of (2.17) are oscillatory 6 Discrete first-order forced oscillation Proof Let {x(n)} be an eventually positive and bounded solution of (1.1) From (2.17) it follows...

Ngày tải lên: 22/06/2014, 22:20

... upper solutions of (2.1)–(2.3) with α(t) ≤ β(t) for all t ∈ J Then, there exists a solution u to (2.1)– (2.3) such that u ∈ [α,β] F M Atici and D C Biles 123 Proof Our proof follows that of Cabada ... there exist a lower solution α and an upper solution β of (3.1)–(3.5) such that α ≤ β on T, then the BVP (3.11)– (3.15) has a solution 128 Dynamic equations with impulse Proof It is not difficult ... respectively, of the problem (3.1)–(3.5) such that α ≤ β on T Then every solution of the BVP (3.11)–(3.15) belongs to the sector [α,β] F M Atici and D C Biles 129 Proof Let y be a solution of (3.11)–(3.15),...

Ngày tải lên: 23/06/2014, 00:20

... than those of Wolff 1.3 Overview of the paper The next two sections of this paper are concerned with reducing the proof of Theorem 1.2 to establishing an existence result for smooth data of small ... the case of smooth, compactly supported data of small norm, namely Proposition 3.1 In Section we present the proof of Proposition 3.1 by the continuity method At the heart of this proof is a recursive ... combination of three terms, each of which takes the form of an operator of type S1,1 acting on du, where any given seminorm of the symbol is bounded by c(m) du L∞ , with c(m) an appropriate power of m.2...

Ngày tải lên: 15/03/2014, 09:20

... end of the introduction we also consider Neumann boundary conditions The statement of the propagation of singularities of solutions has two additional ingredients: locating singularities of a ... the ideal of smoothing operators The topology of Ψbc (X) is given in terms of the conormal seminorms of the Schwartz kernel K of its elements; these seminorms can be stated in terms of the Besov ... quotient of H −1 (X) by the annihilator of H0 (X) In terms of the identification spaces in the penultimate paragraph of Remark 3.1, H −1 (X) = of the H loc ¯ −1 Hloc (X ◦ ) in the notation of [4,...

Ngày tải lên: 15/03/2014, 09:20

Ngày tải lên: 17/03/2014, 14:44

... generator of the satisfying principal filter of the conjunction of such constraints is the unification of the generators of the satisfying principal filters of each of the conjuncts Thus the set of ... restricted to conjunctions of equality constraints (i.e requirements that the value of a path of attributes is equal to a constant or the value of another path) then the set of satisfying attributevalue ... s of '=' d e n o t e identical elements distinct f r o m the d e n o t a t i o n of_ L Even t h o u g h there are infinitely m a n y instances of the s c h e m a in (A5) (since there are infinitely...

Ngày tải lên: 24/03/2014, 02:20

... irrespective of the fact that this sequence is periodic or not, some state will repeat finitely often whereas other will repeat infinitely often The (finite) set of states occurring infinitely often in ... point of view of finite word instead of infinite word reading automata, as for instance in [6,10,11,12,13,14,15] Unfortunately, as opposed to the case of infinite words, the classification theory of ... to the attractive properties of the considered networks, and hence provides a new refined measurement of the computational capabilities of these networks in terms of their attractive behaviours...

Ngày tải lên: 28/04/2014, 10:06

... angular frequency pi k = w k ==2 w c c Wavefront = Line of constant phase Wavelength=shortest distance between adjacent peaks 2D Plane Wave Solution Phase P = Ae = Ae i(k x + k z - (2) wt) x z i(k ... |k|(sin , cos ) k Equation of a line: k r = cnst k is Perpendicular to Wavefront z r k k = |r||k| cos(O) = constant Any pt along line phase is cnst O r (x,z) x 2D Plane Wave Solution Phase P = Ae ... 2pi/T=w=radians/s Phase Time (s) Harmonic Ripples: P = Acos(kx) Wavenumber k=2pi/ Wavelength x Plane Wave Solution Faster Velocities = Stiffer Rocks κ ; c = ρ P = P c Phase=O (1) i(kx-wt) (2) P = Ae dO...

Ngày tải lên: 28/04/2014, 14:16

... single and multiple positive periodic solutions of (1.3) Our aim of this article is to study the existence of periodic solutions for the higherorder difference equations (1.1) using the well-known ... existence of three positive periodic solutions of functional difference equation Georg Math J 18, 39–52 (2011) [9] Gil’, MI, Kang, S, Zhang, G: Positive periodic solutions of abstract difference equations ... −h −1 p−i (Hp x)(n + (i − 1)k), x ∈ X − 1) i=1 Clearly, the fixed point of Tp in X is a periodic solution of (1.1) Proof of Theorem 1.1 Let p = r and Ω = {x ∈ X : m ≤ x(n) ≤ M for n ∈ Z}, then...

Ngày tải lên: 18/06/2014, 22:20