Heat Transfer in a Medium in Which Many Small Particles Are Embedded
|
|
- Cory Scott
- 6 years ago
- Views:
Transcription
1 Math. Model. Nat. Phenom. Vol. 8, No., 23, pp DOI:.5/mmnp/2384 Heat Transfer in a Medium in Which Many Small Particles Are Embedded A. G. Ramm Department of Mathematics Kansas State University, Manhattan, KS , USA Abstract. The heat equation is considered in the complex system consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies a Newtontype boundary condition is imposed. An equation for the limiting field is derived when the characteristic size a of the small bodies tends to zero, their total number N(a) tends to infinity at a suitable rate, and the distance d = d(a) between neighboring small bodies tends to zero a << d. No periodicity is assumed about the distribution of the small bodies. Keywords and phrases: heat transfer, many-body problem Mathematics Subject Classification: 35K2, 35J5, 8M4, 8A2. Introduction Let many small bodies (particles) D m, m M, be distributed in a bounded domain D R 3, diamd m = 2a. The small bodies are distributed according to the law N( ) = a 2 κ N(x)dx[+o()],a. (.) Here D is an arbitrary open subdomain of D, κ (,) is a constant, N(x) is a continuous function, and N( ) is the number of the small bodies D m in. Let us assume that the boundaries of the bodies D m are C 2 smooth. The heat equation can be stated as follows: u t = 2 u+f(x) in R 3 \ M D m := Ω, u t= =, (.2) u N = ζ m u on, m M. (.3) Corresponding author. ramm@math.ksu.edu c EDP Sciences, 23 Article published by EDP Sciences and available at or
2 Here N is the outer unit normal to S := continuous function in D. M, ζ m = h(x m) a κ,x m D m, m M, where h(x) is a The aim of this paper is to establish a method for solving both theoretically and numerically the many-body heat transfer problem, and to derive an equation for the effective (self-consistent) field in the medium in which many small bodies (particles) are embedded. There is a large literature on homogenization theory which deals with similar problems, see []-[4] and the literature therein. In most cases it is assumed in the homogenization literature that the coefficients of the equations are periodic, and there is a small parameter in these coefficients, that the related problems in a periodic cell have discrete spectrum and are selfadjoint, that the elliptic estimates, such as Rellich and Poincare inequalities hold, etc. Our method, which was used in [5]-[9], differs in many respects from the homogenization theory methods developed in the literature: periodicity assumption is not used, the spectrum of the related elliptic problems that we use is continuous, the problems may be non-selfadjoint, and our justification of the limiting equation for the effective field is based on a new technique, namely, on convergence results for collocation methods (see, [7]). The published homogenization techniques are not directly applicable to our problem because of the lack of periodicity. The main results of this paper are: ) Derivation of the linear algebraic system (25) for finding the Laplace transform of the solution to problem (2)-(3), 2) Derivation of the equation (29) for the limiting effective field in the medium when the size of the small bodies tends to zero while the total number of these bodies tends to infinity according to the distribution law (), 3) Derivation of the formula (3) for the average temperature in the limiting medium. In Section 2 a derivation of these results is given, and in Section 3 proofs of some lemmas are given. 2. Derivation of the equation for the limiting effective field Denote Then, (.2) - (.3) imply Let U := U(x,) = e t u(x,t)dt. 2 U +U = f(x) in Ω, (2.) U N = ζ m U on, m M. (2.2) x y g(x,y) := g(x,y,) = e 4π x y, g(x,y)f(y)dy = F(x,). (2.3) R 3 Look for the solution to (2.) - (2.2) of the form U(x,) = F(x,)+ g(x,s)σ m (s)ds, U(x,) := U(x) := U, (2.4) where σ m are unknown and should be found from the boundary conditions (2.2). Equation (2.) is satisfied by U of the form (2.4) for any σ m. To satisfy (2.2) one has to solve equation U e (x) N + A mσ m σ m ζ m U e ζ m T m σ m = on, m M. (2.5) 2 Here U e (x) := U e,m (x) := U(x) g(x,s)σ m (s)ds, (2.6) 94
3 T m σ m = g(s,s )σ m (s )ds g(s,s ), A m σ m = 2 σ m (s )ds, (2.7) N S and the known formula for the outer limiting value on of the normal derivative of a simple layer potential was used. We now apply the ideas and methods for solving many-body scattering problems developed in [5] - [7]. Let us call U e,m the effective (self-consistent) value of U, acting on m-th body. As a, the dependence on m disappears, since g(x,s)σ m (s)ds as a. One has where and J 2 := U(x,) = F(x,)+ g(x,x m )Q m +J 2, x m D m, (2.8) Q m := σ m (s)ds, [g(x,s ) g(x,x m )]σ m (s )ds, J := g(x,x m )Q m. (2.9) We prove (in Section 3) that provided that J 2 << J as a (2.) lim a a =. (2.) d(a) where d(a) = d is the minimal distance between neighboring particles. If (2.) holds, then problem (2.) - (2.2) is solved asymptotically by the formula U(x,) = F(x,)+ g(x,x m )Q m, a, (2.2) provided that asymptotic formulas for Q m, as a, are found. To find formulas for Q m, let us integrate (2.5) over and estimate the order of the terms in the resulting equation as a. We get U e N ds = D m 2 U e dx = O(a 3 ). (2.3) Here we assumed that 2 U e = O(),a. This assumption will be justified in Section 2. A m σ m σ m ds = Q m [+o()], a. (2.4) 2 This relation is justified in Section 2. Furthermore, ζ m U e ds = ζ m U e (x m ) = O(a 2 κ ), a, (2.5) where = O(a 2 ) 95
4 is the surface area of. Finally, ζ m ds g(s,s S )σ m (s )ds = ζ m ds σ m (s ) dsg(s,s ) m Thus, the main term of the asymptotics of Q m is Formulas (2.7) and (2.2) yield and Denote U(x,) = F(x,) U e (x m,) = F(x m,) and write (2.9) as a linear algebraic system where = Q m O(a κ ), a. (2.6) Q m = ζ m U e (x m ). (2.7) g(x,x m )ζ m U e (x m,), (2.8) m m,m = U e (x m,) := U m, F(x m,) := F m, g(x m,x m ) := g mm, U m = F m a 2 κ m m g(x m,x m )ζ m U e (x m,). (2.9) g mm h m c m U m, m M, (2.2) h m = h(x m ), ζ m = h m a κ, c m := a 2. Consider a partition of the bounded domain D, in which the small bodies are distributed, into a union of P << M small nonintersecting cubes p, p P, of side b >> d, b = b(a) as a. Let x p p, p = volume of p. One has a 2 κ M m =,m m g mm h m c m U m = a 2 κ P p =,p p g pp h p c p U p x m p = = p pg pp h p c p U p N(x p ) p [+o()], a. (2.2) Thus, (2.2) yields We have assumed that P U p = F p g pp h p c p N p U p p, p P (2.22) p p,p = h m = h p [+o()], c m = c p [+o()], U m = U p [+o()], a, (2.23) for x m p. This assumption is justified if the functions h(x),u(x,), c(x) = lim x m x,a a 2, 96
5 and N(x) are continuous. The function h(x) and N(x) are continuous by the assumption. The continuity of the U(x,) is proved in Section 3, and the continuity of c(x) is assumed. If all the small bodies are identical, then c(x) = c = const. The sum in the right-hand side of (2.22) is the Riemannian sum for the integral lim a P p =,p p g pp h p c p N(x p )U p = D g(x, y)h(y)c(y)n(y)u(y, )dy (2.24) Therefore, linear algebraic system (2.22) is a collocation method for solving integral equation U(x,) = F(x,) g(x, y)h(y)c(y)n(y)u(y, )dy. (2.25) D Convergence of this method for solving equations with weakly singular kernels is proved in [8]. Applying the operator 2 + and then taking the inverse Laplace transform of (2.25) yields u t = u+f(x) q(x)u, q(x) := h(x)c(x)n(x). (2.26) One concludes that the limiting equation for the temperature contains the term q(x)u. Thus, the embedding of many small particles creates a distribution of source and sink terms in the medium, the distribution of which is described by the term q(x)u. If one solves equation (2.25) for U(x,), or linear algebraic system (2.22) for U p (), then one can Laplaceinvert U(x,) for U(x,t). Numerical methods for Laplace inversion from the real axis are discussed in [9] - []. If one is interested only in the average temperature, one can use the relation T lim u(x,t)dt = lim U(x,) := ψ(x). (2.27) T T Relation (2.27) is proved in Section 3, which holds if the limit on one of its sides exists. The limit on the right-hand side of (2.27) can be calculated by the formula ψ(x) = (I +B) ϕ, ϕ = f(y)dy. (2.28) 4π x y Here, B is the operator Bψ := q(y)ψ(y) dy, q(x) := h(x)c(x)n(x). 4π x y From the physical point of view the function h(x) is non-positive because the flux u of the heat flow is proportional to the temperature u and is directed along the outer normal N: u N = h u, where h = h >. Thus, q. It is proved in [] - [2] that zero is not an eigenvalue of the operator 2 +q(x) provided that q(x) and q = O ( x 2+ǫ ) as x,ǫ >. In our case, q(x) = outside D, so the operator (I +B) exists and is bounded in C(D). Let us formulate the basic result we have proved. Theorem 2.. Assume (.), (2.), and h. Then, there exists the limit U(x,) of U e (x,) as a, U(x,) solves equation (2.25), and there exists the limit (2.27), where ψ(x) is given by formula (2.28). 97
6 3. Proofs of some lemmas Lemma 3.. Assume (2.). Then relation (2.) holds. Proof. One has ( Qm e x x x m ) J,m := g(x,x m )Q = O 4π x x m J 2,m := where x x m 2, and the inequality e Q m x x m, x x m d. (3.) e Sm x xm ( ) ( ) a, 4π x x m max a σ m (s ) ds Qm a O x x m x x m 2, (3.2) max ( e x xm ) e x x m was used. The Q m. In fact, σ m keeps sign on, as follows from equation (2.5) as a. It follows from (3.) - (3.2) that ( ) ( ) J 2,m J,m O a a x x m O <<. (3.3) d From (3.3) by the arguments similar to the given in [3] one obtains (2.). Lemma 3.2. Relation (2.4) holds. Proof. Let us justify relation (2.4). As a, one has It is known (see [5]) that ds e s s N s 4π s s = N s 4π s s + e s s N s 4π s s. (3.4) On the other hand, as a, one has e ds Sm s s 4π s s N s 4π s s σ m(s )ds = σ m (s )ds = Q m. (3.5) σ m (s )ds Q m s s ds Sm e 4π s s = o(q m ). (3.6) The relations (3.5) and (3.6) justify (2.4). Lemma 3.3. Relation (2.27) holds. Proof. Denote Then t t u(t)dt := v(t), ū(σ) := v() = ū(σ) σ dσ e σt u(t)dt. by the properties of the Laplace transform. Assume that the limit v( ) := v exists: lim v(t) = v. (3.7) t 98
7 Then, Indeed, so and (3.7) is verified. One has v = lim lim lim v() = lim Let us check this: lim = lim σū(σ)dσ where we have used the relation σ2dσ =. Alternatively, let σ = γ. Then, σ 2σū(σ)dσ = / e t v(t)dt = lim v(). e t dt =, e t (v(t) v )dt =, = lim ū(). (3.8) σū(σ)dσ / If, then ω =, and if ψ := γ ū(γ ), then References lim ω ω ω σ2σū(σ)dσ = lim σū(σ), (3.9) σ γū( γ )dγ = ω ω γū( )dγ. (3.) γ ψdγ = ψ( ) = lim γ γ ū(γ ) = lim σ σū(σ). (3.) [] V. Jikov, S. Kozlov, O. Oleinik. Homogenization of differential operators and integral functionals. Springer, Berlin, 994. [2] V. Marchenko, E. Khruslov. Homogenization of partial differential equations. Birkhäuser, Boston, 26. [3] D. Gioranescu, P. Donato. An introduction to homogenization. Oxford Univ. Press, Oxford, 999. [4] A. Bensoussan, J.-L.Lions, G. Papanicolau. Asymptotic analysis for periodic structures. AMS Chelsea Publishing, Providence, RI, 2. [5] A. G. Ramm. Wave scattering by small bodies of arbitrary shapes. World Sci. Publishers, Singapore, 25. [6] A.G.Ramm. Inverse problems. Springer, New York, 25. [7] A.G.Ramm. Wave scattering by many small bodies and creating materials with a desired refraction coefficient. Afrika Matematika, 22, No., (2), [8] A.G.Ramm. A collocation method for solving integral equations. Internat. Journ. Comp. Sci and Math., 3, No. 2, (29), [9] A.G.Ramm. Inversion of the Laplace transform from the real axis. Inverse problems, 2, (986), L [] A.G.Ramm, S.Indratno. Inversion of the Laplace transform from the real axis using an adaptive iterative method. Internat. Jour. Math. Math. Sci (IJMMS). Vol. 29, Article 89895, 38 pages. [] A.G.Ramm. Sufficient conditions for zero not to be an eigenvalue of the Schrödinger operator. J. Math. Phys., 28, (987), [2] A.G.Ramm. Conditions for zero not to be an eigenvalue of the Schrödinger operator. J. Math. Phys. 29, (988), [3] A.G.Ramm. Many-body wave scattering by small bodies and applications. J. Math. Phys., 48, No., (27), 35. [4] A.G.Ramm. Wave scattering by many small particles embedded in a medium. Phys. Lett. A, 372/7, (28), [5] A.G.Ramm. A method for creating materials with a desired refraction coefficient. Internat. Journ. Mod. Phys B, 24, No. 27, (2), [6] A.G.Ramm. Materials with a desired refraction coefficient can be created by embedding small particles into a given material. International Journal of Structural Changes in Solids (IJSCS), 2, No. 2, (2), [7] A.G.Ramm. Distribution of particles which produces a smart material. Jour. Stat. Phys., 27, No. 5, (27), [8] A.G.Ramm. Scattering of scalar waves by many small particles. AIP Advances,, (2), [9] A.G.Ramm. Scattering of electromagnetic waves by many small cylinders. Results in Physics,, No., (2),
A method for creating materials with a desired refraction coefficient
This is the author s final, peer-reviewed manuscript as accepted for publication. The publisher-formatted version may be available through the publisher s web site or your institution s library. A method
More informationScattering of electromagnetic waves by many nano-wires
Scattering of electromagnetic waves by many nano-wires A G Ramm Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu Abstract Electromagnetic wave scattering
More informationScattering of scalar waves by many small particles.
This is the author s final, peer-reviewed manuscript as accepted for publication. The publisher-formatted version may be available through the publisher s web site or your institution s library. Scattering
More informationElectromagnetic wave scattering by many small bodies and creating materials with a desired refraction coefficient
168 Int. J. Computing Science and Mathematics, Vol. Electromagnetic wave scattering by many small bodies and creating materials with a desired refraction coefficient A.G. Ramm Mathematics epartment, Kansas
More informationHow to create materials with a desired refraction coefficient?
How to create materials with a desired refraction coefficient? A. G. Ramm Mathematics Department, Kansas State University, Manhattan, KS66506, USA ramm@math.ksu.edu www.math.ksu.edu/ ramm Abstract The
More informationCreating materials with a desired refraction coefficient: numerical experiments
Creating materials with a desired refraction coefficient: numerical experiments Sapto W. Indratno and Alexander G. Ramm Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA
More informationComm. Nonlin. Sci. and Numer. Simul., 12, (2007),
Comm. Nonlin. Sci. and Numer. Simul., 12, (2007), 1390-1394. 1 A Schrödinger singular perturbation problem A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu
More informationarxiv: v1 [math-ph] 20 Jul 2007
Many-body wave scattering by small bodies and applications arxiv:0707.304v [math-ph] 20 Jul 2007 A. G. Ramm (Mathematics epartment, Kansas St. University, Manhattan, KS 66506, USA and TU armstadt, Germany)
More informationA numerical algorithm for solving 3D inverse scattering problem with non-over-determined data
A numerical algorithm for solving 3D inverse scattering problem with non-over-determined data Alexander Ramm, Cong Van Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA ramm@math.ksu.edu;
More informationInverse scattering problem with underdetermined data.
Math. Methods in Natur. Phenom. (MMNP), 9, N5, (2014), 244-253. Inverse scattering problem with underdetermined data. A. G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS 66506-2602,
More informationPhys.Let A. 360, N1, (2006),
Phys.Let A. 360, N1, (006), -5. 1 Completeness of the set of scattering amplitudes A.G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS 66506-60, USA ramm@math.ksu.edu Abstract Let f
More informationAfrika Matematika, 22, N1, (2011),
Afrika Matematika, 22, N1, (2011), 33-55. 1 Wave scattering by small bodies and creating materials with a desired refraction coefficient Alexander G. Ramm epartment of Mathematics Kansas State University,
More informationCreating materials with a desired refraction coefficient: numerical experiments. Sapto W. Indratno and Alexander G. Ramm*
76 Int. J. Computing Science and Mathematics, Vol. 3, Nos. /2, 200 Creating materials with a desired refraction coefficient: numerical experiments Sapto W. Indratno and Alexander G. Ramm* Department of
More informationHow large is the class of operator equations solvable by a DSM Newton-type method?
This is the author s final, peer-reviewed manuscript as accepted for publication. The publisher-formatted version may be available through the publisher s web site or your institution s library. How large
More informationA collocation method for solving some integral equations in distributions
A collocation method for solving some integral equations in distributions Sapto W. Indratno Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA sapto@math.ksu.edu A G Ramm
More informationElectromagnetic wave scattering by small impedance particles of an arbitrary shape
This is the author s final, peer-reviewed manuscript as accepted for publication. The publisher-formatted version may be available through the publisher s web site or your institution s library. Electromagnetic
More informationMany-body wave scattering problems in the case of small scatterers
J Appl Math Comput OI 10.1007/s12190-012-0609-1 PHYICAL CIENCE JAMC Many-body wave scattering problems in the case of small scatterers Alexander G. Ramm Received: 9 August 2012 Korean ociety for Computational
More informationSpectral Properties of Schrödinger-type Operators and Large-time Behavior of the Solutions to the Corresponding Wave Equation
Math. Model. Nat. Phenom. Vol. 8, No., 23,. 27 24 DOI:.5/mmn/2386 Sectral Proerties of Schrödinger-tye Oerators and Large-time Behavior of the Solutions to the Corresonding Wave Equation A.G. Ramm Deartment
More informationCalculation of Electromagnetic Wave Scattering by a Small Impedance Particle of an Arbitrary Shape
Calculation of Electromagnetic Wave cattering by a mall Impedance Particle of an Arbitrary hape A. G. Ramm 1, M. I. Andriychuk 2 1 Department of Mathematics Kansas tate University, Manhattan, K 6656-262,
More informationON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume, Number, Pages S -9939(XX- ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS N. S. HOANG AND A. G. RAMM (Communicated
More information444/,/,/,A.G.Ramm, On a new notion of regularizer, J.Phys A, 36, (2003),
444/,/,/,A.G.Ramm, On a new notion of regularizer, J.Phys A, 36, (2003), 2191-2195 1 On a new notion of regularizer A.G. Ramm LMA/CNRS, 31 Chemin Joseph Aiguier, Marseille 13402, France and Mathematics
More informationScattering by many small particles and creating materials with a desired refraction coefficient
102 Int. J. Computing Science and Mathematics, Vol. 3, Nos. 1/2, 2010 Scattering by many small particles and creating materials with a desired refraction coefficient M.I. Andriychuk Institute of Applied
More informationComputational method for acoustic wave focusing
Int. J. Computing Science and Mathematics Vol. 1 No. 1 7 1 Computational method for acoustic wave focusing A.G. Ramm* Mathematics epartment Kansas State University Manhattan KS 6656-6 USA E-mail: ramm@math.ksu.edu
More informationElectromagnetic wave scattering by small impedance particles of an arbitrary shape
J Appl Math Comput DOI 10.1007/s12190-013-0671-3 ORIGINAL REEARCH JAMC Electromagnetic wave scattering by small impedance particles of an arbitrary shape A.G. Ramm Received: 4 December 2012 Korean ociety
More informationA nonlinear singular perturbation problem
A nonlinear singular perturbation problem arxiv:math-ph/0405001v1 3 May 004 Let A.G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS 66506-60, USA ramm@math.ksu.edu Abstract F(u ε )+ε(u
More informationThe Helmholtz Equation
The Helmholtz Equation Seminar BEM on Wave Scattering Rene Rühr ETH Zürich October 28, 2010 Outline Steklov-Poincare Operator Helmholtz Equation: From the Wave equation to Radiation condition Uniqueness
More informationNONLINEAR DIFFERENTIAL INEQUALITY. 1. Introduction. In this paper the following nonlinear differential inequality
M athematical Inequalities & Applications [2407] First Galley Proofs NONLINEAR DIFFERENTIAL INEQUALITY N. S. HOANG AND A. G. RAMM Abstract. A nonlinear differential inequality is formulated in the paper.
More informationA symmetry problem. R be its complement in R3. We
ANNALE POLONICI MATHEMATICI 92.1 (2007) A symmetry problem by A. G. Ramm (Manhattan, K) Abstract. Consider the Newtonian potential of a homogeneous bounded body R 3 with known constant density and connected
More informationLECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI
LECTURE 5 APPLICATIONS OF BDIE METHOD: ACOUSTIC SCATTERING BY INHOMOGENEOUS ANISOTROPIC OBSTACLES DAVID NATROSHVILI Georgian Technical University Tbilisi, GEORGIA 0-0 1. Formulation of the corresponding
More informationJASSON VINDAS AND RICARDO ESTRADA
A QUICK DISTRIBUTIONAL WAY TO THE PRIME NUMBER THEOREM JASSON VINDAS AND RICARDO ESTRADA Abstract. We use distribution theory (generalized functions) to show the prime number theorem. No tauberian results
More informationAN INVERSE PROBLEM FOR THE WAVE EQUATION WITH A TIME DEPENDENT COEFFICIENT
AN INVERSE PROBLEM FOR THE WAVE EQUATION WITH A TIME DEPENDENT COEFFICIENT Rakesh Department of Mathematics University of Delaware Newark, DE 19716 A.G.Ramm Department of Mathematics Kansas State University
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationA G Ramm, Implicit Function Theorem via the DSM, Nonlinear Analysis: Theory, Methods and Appl., 72, N3-4, (2010),
A G Ramm, Implicit Function Theorem via the DSM, Nonlinear Analysis: Theory, Methods and Appl., 72, N3-4, (21), 1916-1921. 1 Implicit Function Theorem via the DSM A G Ramm Department of Mathematics Kansas
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationAnn. Polon. Math., 95, N1,(2009),
Ann. Polon. Math., 95, N1,(29), 77-93. Email: nguyenhs@math.ksu.edu Corresponding author. Email: ramm@math.ksu.edu 1 Dynamical systems method for solving linear finite-rank operator equations N. S. Hoang
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationDoes negative refraction make a perfect lens?
Does negative refraction make a perfect lens? arxiv:0809.080v1 [cond-mat.mtrl-sci] 1 Sep 008 A. G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-60, USA email: ramm@math.ksu.edu
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationThis article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing
More informationDynamical systems method (DSM) for selfadjoint operators
Dynamical systems method (DSM) for selfadjoint operators A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 6656-262, USA ramm@math.ksu.edu http://www.math.ksu.edu/ ramm Abstract
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS
ON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS PAVEL EXNER 1 AND ANDRII KHRABUSTOVSKYI 2 Abstract. We analyze a family of singular Schrödinger operators describing a
More informationDeterminant of the Schrödinger Operator on a Metric Graph
Contemporary Mathematics Volume 00, XXXX Determinant of the Schrödinger Operator on a Metric Graph Leonid Friedlander Abstract. In the paper, we derive a formula for computing the determinant of a Schrödinger
More informationHomogenization and Multiscale Modeling
Ralph E. Showalter http://www.math.oregonstate.edu/people/view/show Department of Mathematics Oregon State University Multiscale Summer School, August, 2008 DOE 98089 Modeling, Analysis, and Simulation
More informationThis article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing
More informationMidterm Solution
18303 Midterm Solution Problem 1: SLP with mixed boundary conditions Consider the following regular) Sturm-Liouville eigenvalue problem consisting in finding scalars λ and functions v : [0, b] R b > 0),
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationEIGENFUNCTIONS OF DIRAC OPERATORS AT THE THRESHOLD ENERGIES
EIGENFUNCTIONS OF DIRAC OPERATORS AT THE THRESHOLD ENERGIES TOMIO UMEDA Abstract. We show that the eigenspaces of the Dirac operator H = α (D A(x)) + mβ at the threshold energies ±m are coincide with the
More informationMatrix construction: Singular integral contributions
Matrix construction: Singular integral contributions Seminar Boundary Element Methods for Wave Scattering Sophie Haug ETH Zurich November 2010 Outline 1 General concepts in singular integral computation
More informationA Feynman-Kac Path-Integral Implementation for Poisson s Equation
for Poisson s Equation Chi-Ok Hwang and Michael Mascagni Department of Computer Science, Florida State University, 203 Love Building Tallahassee, FL 32306-4530 Abstract. This study presents a Feynman-Kac
More information4 Divergence theorem and its consequences
Tel Aviv University, 205/6 Analysis-IV 65 4 Divergence theorem and its consequences 4a Divergence and flux................. 65 4b Piecewise smooth case............... 67 4c Divergence of gradient: Laplacian........
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationThe first order quasi-linear PDEs
Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.
More informationIntegral Representation Formula, Boundary Integral Operators and Calderón projection
Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton
More informationPropagation of discontinuities in solutions of First Order Partial Differential Equations
Propagation of discontinuities in solutions of First Order Partial Differential Equations Phoolan Prasad Department of Mathematics Indian Institute of Science, Bangalore 560 012 E-mail: prasad@math.iisc.ernet.in
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationCOMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationQ. Zou and A.G. Ramm. Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA.
Computers and Math. with Applic., 21 (1991), 75-80 NUMERICAL SOLUTION OF SOME INVERSE SCATTERING PROBLEMS OF GEOPHYSICS Q. Zou and A.G. Ramm Department of Mathematics, Kansas State University, Manhattan,
More informationEMBEDDING OPERATORS AND BOUNDARY-VALUE PROBLEMS FOR ROUGH DOMAINS
EMBEDDING OPERATORS AND BOUNDARY-VALUE PROBLEMS FOR ROUGH DOMAINS Vladimir M. GOL DSHTEIN Mathematics Department, Ben Gurion University of the Negev P.O.Box 653, Beer Sheva, 84105, Israel, vladimir@bgumail.bgu.ac.il
More informationEIGENFUNCTIONS WITH FEW CRITICAL POINTS DMITRY JAKOBSON & NIKOLAI NADIRASHVILI. Abstract
j. differential geometry 53 (1999) 177-182 EIGENFUNCTIONS WITH FEW CRITICAL POINTS DMITRY JAKOBSON & NIKOLAI NADIRASHVILI Abstract We construct a sequence of eigenfunctions on T 2 of critical points. with
More informationGRONWALL'S INEQUALITY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number I, May 1972 GRONWALL'S INEQUALITY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES DONALD R. SNOW Abstract.
More informationStability of solutions to abstract evolution equations with delay
Stability of solutions to abstract evolution equations with delay A.G. Ramm Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu Abstract An equation u = A(t)u+B(t)F
More informationNEW RESULTS ON TRANSMISSION EIGENVALUES. Fioralba Cakoni. Drossos Gintides
Inverse Problems and Imaging Volume 0, No. 0, 0, 0 Web site: http://www.aimsciences.org NEW RESULTS ON TRANSMISSION EIGENVALUES Fioralba Cakoni epartment of Mathematical Sciences University of elaware
More informationNOTES ON NEWLANDER-NIRENBERG THEOREM XU WANG
NOTES ON NEWLANDER-NIRENBERG THEOREM XU WANG Abstract. In this short note we shall recall the classical Newler-Nirenberg theorem its vector bundle version. We shall also recall an L 2 -Hörmer-proof given
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationReal Analysis Qualifying Exam May 14th 2016
Real Analysis Qualifying Exam May 4th 26 Solve 8 out of 2 problems. () Prove the Banach contraction principle: Let T be a mapping from a complete metric space X into itself such that d(tx,ty) apple qd(x,
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationPower Series and Analytic Function
Dr Mansoor Alshehri King Saud University MATH204-Differential Equations Center of Excellence in Learning and Teaching 1 / 21 Some Reviews of Power Series Differentiation and Integration of a Power Series
More informationNonlinear Control. Nonlinear Control Lecture # 3 Stability of Equilibrium Points
Nonlinear Control Lecture # 3 Stability of Equilibrium Points The Invariance Principle Definitions Let x(t) be a solution of ẋ = f(x) A point p is a positive limit point of x(t) if there is a sequence
More informationMATH 819 FALL We considered solutions of this equation on the domain Ū, where
MATH 89 FALL. The D linear wave equation weak solutions We have considered the initial value problem for the wave equation in one space dimension: (a) (b) (c) u tt u xx = f(x, t) u(x, ) = g(x), u t (x,
More informationThe Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge
The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge Vladimir Kozlov (Linköping University, Sweden) 2010 joint work with A.Nazarov Lu t u a ij
More informationMath 273, Final Exam Solutions
Math 273, Final Exam Solutions 1. Find the solution of the differential equation y = y +e x that satisfies the condition y(x) 0 as x +. SOLUTION: y = y H + y P where y H = ce x is a solution of the homogeneous
More informationOn the spectrum of a nontypical eigenvalue problem
Electronic Journal of Qualitative Theory of Differential Equations 18, No. 87, 1 1; https://doi.org/1.143/ejqtde.18.1.87 www.math.u-szeged.hu/ejqtde/ On the spectrum of a nontypical eigenvalue problem
More informationComments on the letter of P.Sabatier
Comments on the letter of P.Sabatier ALEXANDER G. RAMM Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu Aug. 22, 2003. Comments on the letter of P.Sabatier,
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist
Assn 3.1-3.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it exists. 1) Find: lim x -1 6x + 5 5x - 6 A) -11 B) - 1 11 C)
More informationSPECTRAL PROPERTIES OF THE SIMPLE LAYER POTENTIAL TYPE OPERATORS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43, Number 3, 2013 SPECTRAL PROPERTIES OF THE SIMPLE LAYER POTENTIAL TYPE OPERATORS MILUTIN R. OSTANIĆ ABSTRACT. We establish the exact asymptotical behavior
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS INVOLVING A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL
Electronic Journal of Differential Equations, Vol. 217 (217), No. 289, pp. 1 6. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationarxiv:math/ v2 [math.ap] 3 Oct 2006
THE TAYLOR SERIES OF THE GAUSSIAN KERNEL arxiv:math/0606035v2 [math.ap] 3 Oct 2006 L. ESCAURIAZA From some people one can learn more than mathematics Abstract. We describe a formula for the Taylor series
More informationLACK OF HÖLDER REGULARITY OF THE FLOW FOR 2D EULER EQUATIONS WITH UNBOUNDED VORTICITY. 1. Introduction
LACK OF HÖLDER REGULARITY OF THE FLOW FOR 2D EULER EQUATIONS WITH UNBOUNDED VORTICITY JAMES P. KELLIHER Abstract. We construct a class of examples of initial vorticities for which the solution to the Euler
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationNumerical methods for a fractional diffusion/anti-diffusion equation
Numerical methods for a fractional diffusion/anti-diffusion equation Afaf Bouharguane Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux 1, France Berlin, November 2012 Afaf Bouharguane Numerical
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Spring 208 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationEuler Equations: local existence
Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u
More informationand finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationMATH 425, FINAL EXAM SOLUTIONS
MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u
More informationA Characterization of Einstein Manifolds
Int. J. Contemp. Math. Sciences, Vol. 7, 212, no. 32, 1591-1595 A Characterization of Einstein Manifolds Simão Stelmastchuk Universidade Estadual do Paraná União da Vitória, Paraná, Brasil, CEP: 846- simnaos@gmail.com
More informationA PROPERTY OF SOBOLEV SPACES ON COMPLETE RIEMANNIAN MANIFOLDS
Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A PROPERTY
More informationMATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
More informationMan Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No. 4, December 26 GIRSANOV THEOREM FOR GAUSSIAN PROCESS WITH INDEPENDENT INCREMENTS Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim *** Abstract.
More informationKANSAS STATE UNIVERSITY Manhattan, Kansas
NUMERICAL METHODS FOR SOLVING WAVE SCATTERING by NHAN THANH TRAN B.S., University of Science, Vietnam, 2003 M.S., McNeese State University, 2010 AN ABSTRACT OF A DISSERTATION submitted in partial fulfillment
More informationFinding discontinuities of piecewise-smooth functions
Finding discontinuities of piecewise-smooth functions A.G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu Abstract Formulas for stable differentiation
More information