maxwell s equations boundary conditions potentials and power and energy

... Contents | TIME-VARYING FIELDS AND MAXWELL' S EQUATIONS We now wish to define suitable time-varying potentials which are consistent with the above expressions when only static charges and direct ... quantities, and their results therefore are expressed in terms of integral relationships A differential equation always represents a theory Let us now collect the integral forms of Maxwell' s equations ... B Á dS ˆ …36† These four integral equations enable us to find the boundary conditions on B, D, H, and E which are necessary to evaluate the constants obtained in solving Maxwell' s equations in...

Ngày tải lên: 07/08/2014, 20:23

... free space is transverse This is exactly what is observed in the Fresnel–Arago interference equations Thus, in Maxwell s theory this result is an immediate consequence of his equations, whereas ... the same Maxwell s equations showed that the wave equation for optics, if his theory was correct, was no longer a hypothesis but rested on firm experimental and theoretical foundations The association ... equation (which shall be discussed shortly) and Newton s second law of motion, these equations provide a complete description of the classical dynamics of interacting charged particles and electromagnetic...

Ngày tải lên: 19/04/2016, 11:40

... g ij (25) (26) where γ > is the constant with dimension of the squared mass, α = is the dimensionless constant which specifies the rigidity of the string world sheet is the Laplace-Beltrami operator ... of Science, Mathematics - Physics 24 (2008) 111-118 118 Conclusion The equations of motion, of the boundary conditions and of the energy - momentum for the classical rigid string are reconserved ... terms Acknowledgements The author would like to thank Profs B.M.Barbashov,V.V.Nesterenko for useful discussions This work was partly supported by Vietnam National Research Programme in National Science...

Ngày tải lên: 05/03/2014, 14:20

... X\{0} and s large enough, E(sw) ≤ d1 s p+ −d3 s µ −d4 s µ , thus by (H3 ), We conclude that E(sw) → −∞ as s → +∞ So by the Mountain Pass lemma this theorem is proved By the symmetric Mountain Pass ... where the energy functional associated with problem (P ) is coercive And in Section 4, we consider the case where the energy functional possesses the Mountain Pass geometry Preliminaries In order ... and hence E is coercive Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some u0 ∈ X In this case E is differentiable at u0 , then u0 is a solution...

Ngày tải lên: 20/06/2014, 21:20

... determinants and linear functionals We construct a special basis of the solutions in Section and introduce some functions that are independent of this basis The expression of the solution to the second-order ... problems, conditions for positiveness of Green s functions, and solutions with various BCs, for example, NBCs The structure of the paper is as follows In Section 2, we review the properties of functional ... one-dimensional three-pointwise equations are described Also, modern economic direct methods for solving Poisson difference equations in a rectangle with boundary conditions of various types are stated...

Ngày tải lên: 21/06/2014, 11:20

... positive solutions Discussion In this section, we offer some interesting discussions associated with problem 1.6 Discussion Generally, it is difficult to obtain the upper and lower bounds of positive ... ν∗ sg s ds, sh s ds 4.4 ∗ Lemma 4.1 If H1 and H2 hold, then BVP 4.2 has a unique solution x and x can be expressed in the form x t H ∗ t, s f s, y s ds, 4.5 where H ∗ t, s G t, s t 1− sh s ds ... have g s y s ds 1− 1 g s ds 0 G s, r g r dr φq x s ds m g s k 1 y t ds , G s, tk Ik y tk m G t, s φq x s ds 2.27 G t, tk Ik y tk k 1 1−μ 1 0 G s, r g r dr φq x s ds m g s G s, tk Ik y tk ds k...

Ngày tải lên: 21/06/2014, 16:20

... plantation estates Smallscale rubber farmers and price fluctuations Addressing CSR issues in supply chains 7a Mattress brands 7b Rubber glove brands 7c Rubber boots brands 7d Condom brands ... companies adress CSR in their supply chains: - None of the companies – besides Ikea – were found to adress rubber producers, such as plantation estates, in CSR policies Ikea adress CSR issues according ... as well as the focus on these challenges by some of the rubber brands used by most consumers and in the Danish health sector Methodology This research is based on desk as well as field investigations...

Ngày tải lên: 21/02/2014, 04:20

... SGML that stresses content over design XSD XSD stands for XML Schema Definition (see Schema, earlier in this list) XSL Means eXtensible Stylesheet Language This is a language used to style XML ... coolest tricks of XML, you can use entities to insert names, special typesetting characters or symbols, or long passages of text and data stored in separate files Some entities are standard (those ... make things here as fun and painless as possible—but yes, you will see a bit of code in this chapter XML Basics for Designers XML stands for eXtensible Markup Language A markup language is a technique...

Ngày tải lên: 14/03/2014, 23:20

... relations, and then derives explicit expressions for the quasi-TE and quasi-TM modes It shows formulations of Ex and Hy expressions for the quasi-TE (transverse electric) mode and Ey and Hx expressions ... on equations is risky unless one knows the approximations used in deriving those equations, the errors due to those approximations, and the stability of the solutions Mastering several kinds of ... stabilities of the solutions are discussed as clearly and concisely as possible Someone using this book as a reference should be able to understand the papers in the ®eld, develop programs, and even...

Ngày tải lên: 17/03/2014, 14:41

... Perlmutter and Ratiu [1998] As these and related references show, the Lie–Poisson equations apply to a surprisingly wide variety of systems such as the heavy top, compressible flow, stratified incompressible ... understood In addition, several classical cases of reduction (using conservation laws and/ or symmetry to create smaller dimensional phase spaces), such as the elimination of cyclic variables as ... Phys., 78, 455–478 Arms, J M., J E Marsden and V Moncrief [1982], The structure of the space solutions of Einstein s equations: II Several Killings fields and the Einstein–Yang–Mills equations, ...

Ngày tải lên: 24/04/2014, 16:49

Ngày tải lên: 12/06/2014, 16:17

... 0) | ds η + |ν1 |t (η − s) 2α s |f (s, x (s) , (φx) (s) , (ψx) (s) ) − f (s, 0, 0, 0) (2α − 1) α−1 + |f (s, 0, 0, 0) | ds ⎛ t (t − s) α−1 ≤ sup ⎝ (L1 (s) |x (s) | + L2 (s) | (φx) (s) | + L3 (s) | ... u is a solution of (1.1) and (1.2)-(1.3), then t u(t) = (t − s) α−1 f (s, u (s) , (φu) (s) , (ψu) (s) ) ds (α) η + ν1 t α−1 (η − s) 2α−2 f (s, u (s) , (φu) (s) , (ψu) (s) ) ds, (2α − 1) Ahmad and ... |f s, u (s) , (φu) (s) , (ψu) (s) − f (s, υ (s) , (φv) (s) , (ψv) (s) ) |ds (2α − 1) (t − s) α−1 (L1 (s) |u − v| + L2 (s) |φv| + L3 (s) |ψu − ψv|) ds (α) (η − s) 2α−2 | L1 (s) |u − v| + L2 (s) |φu −...

Ngày tải lên: 20/06/2014, 22:20

... Fuzzy Sets Syst 159, 730–738 (2008) doi:10.1016/j.fss.2007.07.011 Mirmostafee, AK, Moslehian, MS: Fuzzy versions of Hyers-Ulam-Rassias theorem Fuzzy Sets Syst 159, 720–729 (2008) doi:10.1016/j.fss.2007.09.016 ... Trans Am Math Soc 347, 3111–3119 (1995) doi:10.2307/2154775 Hyers, DH, Isac, G, Rassias, ThM: Stability of Functional Equations in Several Variables Birkhäuser, Basel (1998) Hyers, DH, Isac, ... Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality Math Inequal Appl 4, 93–118 (2001) Jung, S- M: Hyers-Ulam-Rassias stability of Jensen s equation and its application Proc Am Math Soc...

Ngày tải lên: 20/06/2014, 22:20

... Cite this article as: Tahamtani and Peyravi: Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions Boundary Value Problems 2011 2011:19 Submit your ... of Horgan and Knowles [1] The interested reader is referred to the papers [2-9] and the reviews by Horgan and Knowles [1,10,11] The energy method is widely used to study such results Spatial growth ... recent years, much attention has been directed to the study of spatial behavior of solutions of partial differential equations and systems The history and development of this question is explained...

Ngày tải lên: 21/06/2014, 00:20

... ideal solids and liquids and are named viscoelastic materials The basic models of such materials are based on combinations of springs and dashpots 2, 16, 54 As an illustration, let us consider the ... combination of a spring and a dashpot in series In this case, the stresses are equal for both structural parts while the strains add This leads to the classical Maxwell model which is described by ... calculus and the classical and quantum mechanics 44, 45 This is also the approach of the works of Rabei and coworkers 46 Fractional Calculus and Fractals Fractals and fractional calculus generate...

Ngày tải lên: 21/06/2014, 05:20

... one says that J satisfies the P S c condition If J satisfies P S c condition for all c ∈ R, one says that J satisfies the P S condition Lemma 2.2 see Assume that u and u are, respectively, lower and ... Mountain pass theorem in half-order intervals, sup-solutions case see v0 < v1 is Suppose that J satisfies (J1 )–(J3 ) v1 < v2 is a pair of strict supersolution of ∇J a subsolution of ∇J Suppose that ... The assumption is not essential but is assumed for simplicity If such m does not exist then we can approximate f by a sequence of functions so that m as above exists, and obtain the solutions by...

Ngày tải lên: 21/06/2014, 07:20

... 2.2 Suppose that u x, t is a weak solution of problem 1.1 – 1.4 and g s satisfies: s sg s ≥ KG s , where K > 2, G s S1 1−p /4 4/ p−1 E0 e where C1 u0 mS1 S1 u1 S2 −2 S1 b/a u0 g ρ dρ, G s ≥ β |s| p ... u0 ∈ H s 1/2 S1 , u1 ∈ H s S1 , s > 1, and g ∈ C and that the problem 1.1 – 1.4 possesses a unique local weak solution 2, 3, that is, u is in the class u ∈ L∞ 0, T ; H s ut ∈ L∞ 0, T, H s S1 , ... of the usually space H Ω , H s S , Lp Ω , and Lp S can be found in 32 and the norm of L2 S is denoted by • S Boundary Value Problems Blowup of the Solutions In this paper, we always assume that...

Ngày tải lên: 21/06/2014, 11:20

... possesses at least n distinct pairs of nontrivial critical points Then, problem 3.19 has at least n distinct pairs of nontrivial classical solutions, that is, problem 1.1 has at least n distinct ... possesses at least n distinct pairs of nontrivial critical points Then, problem 3.35 has at least n distinct pairs of nontrivial classical solutions, that is, problem 1.1 has at least n distinct ... uk is bounded and ϕ uk → as k → ∞ possesses a convergent subsequence in E, we say that ϕ satisfies the Palais-Smale condition Let E be a real Banach space Define the set Σ {A | A ⊂ E \ {θ} as symmetric...

Ngày tải lên: 21/06/2014, 11:20

... ak h s bk ak bk ak h s z s ds ds Gk s, τ σ τ dτ ds h s z s ds, 2.19 therefore, bk ak h s z s ds 1− bk ak h s ds r3 bk ak bk ak r4 − r3 bk − ak h s ds h s bk ak bk ak s − ak h s ds 2.20 Gk s, τ ... t0 − s f s, − s ds < ∞ 3.33 Similarly, by integration of 3.29 , we get t0 sf s, − s ds < ∞ 3.34 Equations 3.33 and 3.34 imply that s − s f s, − s ds < ∞ 3.35 Since x t is a C2 0, positive solution ... g s x s ds, 4.9 ϕp x h s ϕp x s ds, x 0 Define the linear operator A1 as follows: A1 x t G t, s x s ds t 1− sg s ds G s, τ g τ x s dτ ds 4.10 Boundary Value Problems 21 If x t is a C2 0, positive...

Ngày tải lên: 21/06/2014, 11:20

... ≤ η s Γ q |a| ≤ t ds f s, 0, 0, f s, x s , φx s , ψx s −f s, 0, 0, L2 s φx s L3 s f s, 0, 0, ds ds M ds ψx s /2 tq−1 /2 q−1 L1 s x s t − s q−1 L1 s x s Γ q aΓ q − f s, 0, 0, f s, x s , φx s , ... q ds L2 s γ0 L2 s x s /2 tq−1 η q−1 Γ q φx s /2 L3 s φx s δ0 L s x s M ds ψx s L3 s ψx s M ds M ds Advances in Difference Equations η q−1 η s × Γ q |a| ≤ I q L1 t × L1 s x s γ0 L2 s x s − s q−1 ... operators Θ1 and Θ2 on Br as Θ1 x t − t Θ2 x t t − s q−1 f s, x s , φx s , ψx s ds, Γ q Γ q η q−1 Γ q aΓ q × /2 tq−1 η η s Γ q a /2 3.10 q−1 f s, x s , φx s , ψx s ds − s q−1 /2 f s, x s , φx s ,...

Ngày tải lên: 21/06/2014, 16:20