exact solutions of some dual equations of the first kind

... the difference of the time delays t = tq-tT These studies have pointed out that the usefulness of the Cattaneo-Vernotte and DPL exact models is limited In this study, by means of the method of ... precedence of the heat flux vector to the temperature gradient at all In fact, whether the heat flux vector precedes the temperature gradient or not depends on the combined effects of the thermal ... where a[m2.s-1] is the thermal diffusivity of the medium, and t = tq-tT is the difference of the phase lags Equation shows explicitly that the DPL and SPL models, both in their exact form, are entirely...

Ngày tải lên: 21/06/2014, 04:20

... C = const On the Smoothness of Solutions 145 For j = we obtain the statement Propostion 2.2 is proved We can now state our theorem on the smoothness of generalized solutions of the problem (2.1) ... ∂ k apq q D ∂tk By Theorem 2.1 the function v = uts still satisfies the boundary conditions Therefore from inductive hypothesis and by repeating arguments of the proof in the case s = 0, we obtain ... k=0 l L∞ (0,∞;H0 (Ω)) (2.17) The function ϕ1 u = (1−ϕ0 )u equals in some neighborhood of the conical point We can apply theorems on the smoothness of solutions of elliptic problems in a 2m+l...

Ngày tải lên: 06/08/2014, 05:20

... R(π,f ) = ∅, let φ((π, f )) = (π, f ) Otherwise, let (i, j) ∈ R(π,f ) be minimal under the lexicographic ordering of R(π,f ) Let π = (i, j) ◦ π, the product of the transposition (i, j) and π in Sn ... 49 of Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge, 1997 With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original the electronic journal of ... Janet Beissinger, who was the first to explore [1] combinatorial proofs using colored permutations, for their assistance References [1] Janet Beissinger Colorful proofs of generating function identities...

Ngày tải lên: 08/08/2014, 01:20

... but also the fact that they are themselves solutions to a similar system of equations This allowed us to improve the exponent s, needed √ in the proof of well-posedness of equations of type (6),8 ... of the Main Theorem with the exception of those results which concern the asymptotic properties of the Ricci coefficients (the Asymptotics Theorem), and the straightforward modifications of the ... desired in the proof of (93) The remaining term on the left-hand side of (93) is easier to treat Proof of the L2 − L∞ decay estimate; Theorem 2.1 In this section we rely on the Boundedness Theorem...

Ngày tải lên: 16/02/2014, 05:20

... Exact Solutions of Einstein’s Field Equations A revised edition of the now classic text, Exact Solutions of Einstein’s Field Equations gives a unique survey of the known solutions of Einstein’s ... vacuum solutions 29.1 Twisting vacuum solutions – the field equations 29.1.1 The structure of the field equations 29.1.2 The integration of the main equations 29.1.3 The remaining field equations ... surprise for the reader may lie in the enormous number of known exact solutions Those who not work in the field often suppose that the 1.2 The development of the subject intractability of the full...

Ngày tải lên: 24/04/2014, 17:07

... Exact Solutions of Einstein’s Field Equations A revised edition of the now classic text, Exact Solutions of Einstein’s Field Equations gives a unique survey of the known solutions of Einstein’s ... vacuum solutions 29.1 Twisting vacuum solutions – the field equations 29.1.1 The structure of the field equations 29.1.2 The integration of the main equations 29.1.3 The remaining field equations ... surprise for the reader may lie in the enormous number of known exact solutions Those who not work in the field often suppose that the 1.2 The development of the subject intractability of the full...

Ngày tải lên: 24/04/2014, 17:07

... behavior of solutions of the system of difference equations xn+1 = a + xn c + yn e + zn , yn+1 = , zn+1 = b + yn d + zn f + xn In [8], Özban studied the positive solutions of the system of rational ... rational difference equations xn+1 = yn−k , yn+1 = yn xn−m yn−m−k In [9], Zhang et al investigated the behavior of the positive solutions of the system of the difference equations xn = A + yn−p ... behavior of a three-dimensional linear fractional system of difference equations J Math Anal Appl 310, 673–689 (2005) Özban, AY: On the positive solutions of the system of rational difference equations...

Ngày tải lên: 20/06/2014, 22:20

... aspects of these equations are yet to be explored For some recent work on equations of fractional order with deviated arguments, see [46-48] and the references therein In this article, we consider the ... mentioning that the conditions of our theorems are easily to verify, so they are applicable to a variety of problems, see Examples 4.1 and 4.2 The proof of our main results is based upon the following ... t∈[η,1] β(1 − η) , η} − βη Proof Note that u“ (t) ≤ 0, by applying the concavity of u, the proof is easy So we omit it where γ = min{βη, Proofs of main theorems Define the operator T : C[0, 1] ®...

Ngày tải lên: 21/06/2014, 03:20

... results of If we put σ these theorems, the same results as Theorem in are obtained in We will prove Theorem 1.1 a and b in Sections and 4, respectively The proof of Theorem 1.2 is included in the ... 373–390, 1993 R Suzuki, “Existence and nonexistence of global solutions of quasilinear parabolic equations, ” Journal of the Mathematical Society of Japan, vol 54, no 4, pp 747–792, 2002 C Bandle, ... is called the critical exponent On the other hand, 6–9 and so on study the inhomogeneous equations i.e., f x / ≡ in 1.1 Bandle et al study the case m 1, σ 0, and Zeng and Zhang study the case...

Ngày tải lên: 21/06/2014, 11:20

... Difference Equations 3.2 The Error Bound for the Asymptotic Expansion of y2 (n) Now we estimate the error bound of the asymptotic expansion of the linear independent solution y2 n to the original ... problem of obtaining error bounds for these asymptotic solutions to 1.2 is still open The purpose of this and the next paper Error bounds for asymptotic solutions of second-order linear difference equations ... values of parameters The first case is devoted to the situation when k > 0, and in the second case as k < where k 2p − q The whole proof of the result is too long to understand, so we divide the...

Ngày tải lên: 21/06/2014, 11:20

... By this the proofs of the part of the results (which require the rather long proofs) are given in Section In Section 4, the main results are formulated, in Sections 5–7 the proofs of the main ... respect to γk ¯ The definition of the L-condensing map is given in [3] The proof of theorem repeats the proof of Theorem 2.2 in [21] on the strength of ¯ ¯ Lemmas 5.6–5.8 and the inequality u ... [17]), and γn is the operator of taking of the normal component of a trace on the boundary of a function defined on Ω0 The operator γn 236 On weak solutions of the equations of motion 1/2 is...

Ngày tải lên: 23/06/2014, 00:20

... note that by virtue of (3.10), the function u is the solution of (3.12), and according to the unique solvability of that equation, u = v ≥ in the domain DT Thus the proof of Lemma 3.3 is complete ... C(DT ) (3.19) Therefore the series v = limm→ ∞ vm = v0 + ∞=0 wm converges in the class C(DT ) and its m sum is the solution of (3.12) The uniqueness of the solution (3.12) in the space C(DT ... ), and F ≥ Then if u ∈ C (DT ) is the classical solution of problem (2.1)-(2.2), then u ≥ in the domain DT ◦ Proof If u ∈ C (DT ) is the classical solution of problem (2.1)-(2.2), then u ∈ C...

Ngày tải lên: 23/06/2014, 00:20

... 8 Numerical Solutions of the Black Scholes Equation The left-hand side of the heat equation, on the other hand, cannot be unambiguously approximated The following are some of the more common ... exponentials of the first column Note that the S values in this grid correspond to those of the Jarrow– Rudd scheme of Figure 7.4 If the reader is unsure of the reason for this, he will find the answer ... conditions) and the values at n+1 n the top and bottom edges (boundary conditions); the um−1 um−1 solution of the problem is the series of values at the right-hand edge In order to find these right-hand...

Ngày tải lên: 25/10/2013, 19:20

... classes of exact solutions of the original system of PDEs (Sachdev and Reddy (1982)) There is yet another way of extending the class of similarity solutions This is to embed the similarity solutions, ... similarity solutions exist, their existence theory greatly assists in the understanding of the original PDE system These solutions also help in the quantitative estimation of how the solutions of certain ... explained the nature of self-similar solutions of the first and second kind More importantly, Barenblatt brought out manifestly the role of these solutions as intermediate asymptotics; these solutions...

Ngày tải lên: 24/04/2014, 17:09

... yields the uniqueness of the solution of problems (12), (17), and (18) What is the structure of the solutions of system (12) that increase slower than r-2n-1 in the case where n does not satisfy the ... const on the interval (0, R) Then, the condition qn1j (R) = yields the identity ˜ ˜ qn1j (r) ≡ because of the continuity of the function qn1j on (0, R) In such a case, the ˜ n satisfy the equation ... coefficients Bi(x) have the main influence to the asymptotic of the solutions of system (1), then the last requirement is dispensable [8,9]) Therefore, it is important to consider the case where C(0)...

Ngày tải lên: 21/06/2014, 00:20

... 2.7 The proof of Lemma 2.3 is completed Main Results In this section, we investigate the boundedness of solutions of 1.1 Let q > p > 1, and let {xn }∞ −1 be a positive solution of 1.1 with the ... for any n ≥ Let S be the set of all initial values x−1 , x0 ∈ 0, ∞ × 0, ∞ such that the positive solutions {xn }∞ −1 of 1.1 are bounded Then we have the following theorem n Theorem 3.6 S ∞ n Qn ... }n∞ of equation xn −1 F xn , xn−1 , {x2n }∞ and {x2n−1 }∞ exactly one of the following n n They are both monotonically increasing They are both monotonically decreasing Eventually, one of them...

Ngày tải lên: 21/06/2014, 20:20

... unique solution f ∈ A N1 , N2 Proof The existence of 1.2 in A N1 , N2 is given by Theorem 3.1, from the proof of Theorem 3.1, we see that A N1 , N2 is a closed subset of C1 I, I , by 3.12 and P3 , ... Acta Mathematica W N Zhang, “Stability of the solution of the iterated equation n λi f i x i Scientia, vol 8, no 4, pp 421–424, 1988 W N Zhang, “Discussion on the differentiable solutions of the ... C1 I, I 2.1 is a complete space The following lemmas are useful, and the methods of proof are similar to those of paper , but the conditions are weaker than those of Lemma 2.1 Suppose that ϕ ∈...

Ngày tải lên: 22/06/2014, 11:20

... Brzdek, Subgroups of the group Zn and a generalization of the Gołab-Schinzel functional equation, ¸ Aequationes Mathematicae 43 (1992), no 1, 59–71 , Some remarks on solutions of the functional ... Approximate solutions of the Gołab-Schinzel equation ¸ Remark 2.7 Theorem 2.5 remains true, if instead of the hemicontinuity at the origin, we assume that for every x ∈ X \ f −1 ({0}), there exists ... x + f (x)k y (2.17) Approximate solutions of the Gołab-Schinzel equation ¸ Note that in view of (2.1) and (2.13), for every x, y ∈ X, the right-hand side of the last equality is bounded Thus...

Ngày tải lên: 22/06/2014, 22:20

... For the former, 83.3% of the teacher often grouped students of the same sex while only 20% of the students chose it and for the latter, group students of different sexes was the choice of 80% of ... (1986:161), at the level of language theory, CLT has a rich theoretical base and some of the characteristics of the communicative view of language are: - Language is a system for the expression of meaning ... skill for the first- year students is The number of the first- year English major students is estimated at more than 300 They come from various parts of the country Their levels of English proficiency...

Ngày tải lên: 07/11/2012, 14:50