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Nonlocal Vector Calculus by Fahad Hamoud Almutairi B.E., Kuwait University, 2013 a REPORT Submitted in Partial Fulfillment of Th

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Nonlocal Vector Calculus by Fahad Hamoud Almutairi B.E., Kuwait University, 2013 a REPORT Submitted in Partial Fulfillment of Th Nonlocal Vector Calculus by Fahad Hamoud Almutairi B.E., Kuwait University, 2013 A REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mathematics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2018 Approved by: Major Professor Bacim Alali Abstract Nonlocal vector calculus, introduced in4;6, generalizes differential operators' calculus to nonlocal calculus of integral operators. Nonlocal vector calculus has been applied to many fields including peridynamics, nonlocal diffusion, and image analysis1{3;5;7. In this report, we present a vector calculus for nonlocal operators such as a nonlocal divergence, a non- local gradient, and a nonlocal Laplacian. In Chapter 1, we review the local (differential) divergence, gradient, and Laplacian operators. In addition, we discuss their adjoints, the divergence theorem, Green's identities, and integration by parts. In Chapter 2, we define nonlocal analogues of the divergence and gradient operators, and derive the corresponding adjoint operators. In Chapter 3, we present a nonlocal divergence theorem, nonlocal Green's identities, and integration by parts for nonlocal operators. In Chapter 4, we establish a con- nection between the local and nonlocal operators. In particular, we show that, for specific integral kernels, the nonlocal operators converge to their local counterparts in the limit of vanishing nonlocality. Table of Contents List of Figures . iv Introduction . .1 1 Local Operators . .2 1.1 The divergence theorem . .3 1.2 Adjoints for the divergence and the gradient operators . .3 1.3 Integration by parts formula . .4 1.4 Green's identities for local operators . .5 2 Nonlocal Operators . .7 2.1 Overview . .7 2.2 Nonlocal operators . .9 2.3 Free space nonlocal Calculus theorem . 10 2.4 Adjoints of nonlocal operators . 11 3 Nonlocal Interaction Operators . 14 3.1 Interaction domain . 14 3.2 Nonlocal interaction operators . 15 3.3 A nonlocal integral theorem . 16 3.4 Nonlocal integration by parts formulas . 17 3.5 Nonlocal Green's identities . 19 4 A Connection Between Local and Nonlocal Operators . 21 4.1 Nonlocal Laplacian operator . 21 References . 25 iii List of Figures 4 3.1 A domain Ω and different possibilities for its interaction domain ΩI ..... 15 iv Introduction In this report, we present a vector calculus for nonlocal operators such as a nonlocal di- vergence, a nonlocal gradient, and a nonlocal Laplacian. We define nonlocal analogues of divergence and gradient and derive the corresponding adjoint operators; this will be the topic of Chapter 2. In Chapter 3, we present a nonlocal divergence theorem, Green's identities, and integration by parts formula for nonlocal operators. In Chapter 1, we review the local divergence and gradient operators and their adjoints, the local divergence theorem, the local Green's identities and integration by parts. In Chapter 4, we establish a connection between local and nonlocal operators. In particular, we show that, for an appropriate integral kernel, the nonlocal Laplacian operator converges to the local Laplacian operator in the limit of vanishing nonlocality. 1 Chapter 1 Local Operators We review some definitions of classical operators in Cartesian coordinates in Rn. Local divergence operator Let v be a differentiable vector field v : Rn ! Rk. Then dv dv dv div v = r · v = 1 + 2 + ::: + k : dx1 dx2 dxk Local gradient operator df df df n grad f = rf = ( ; ;:::; ); where f : R ! R is differentiable. dx1 dx2 dxn Local Laplacian operator df df d2f d2f ∆f = r · rf = r · ( ;:::; ) = 2 + ::: + 2 ; dx1 dxn dx1 dxn where f; g are twice differentiable. A product rule The divergence of the product of a vector field v and a scalar field u satisfies r · (uv) = ur · v + v · ru (1.1) 2 1.1 The divergence theorem Let v be a differentiable vector field, Ω a bounded domain, and n a unit normal vector. Then Z Z r · v dx = v · n ds: Ω @Ω In Rn and for v compactly supported, the divergence theorem is given by Z r · v dx = 0: Rn 1.2 Adjoints for the divergence and the gradient oper- ators Let V and U be Hilbert spaces. Then we recall that the adjoint T ∗ of a linear operator T : V ! U satisfies ∗ < T v; u >U =< v; T u >V ; for all v 2 Dom(T ) and u 2 Dom(T ∗). Next we show the following: Proposition 1.1. The adjoints of the gradient and divergence operators are given by (I) (∇·)∗ = −∇ (II) (r)∗ = −∇· Proof. Let v be a differentiable vector field and u be a differentiable compactly supported scalar field. For (I), we want to show that < r · v; u >=< v; −∇u >. Using the identity (1.1) along with the divergence theorem Z Z < r · v; u > = (r · v) u dx = − v · ru dx n n ZR R = v · (−∇u) dx = < v; −∇u >; Rn 3 and hence (∇·)∗ = −∇ as required. For (II), we want to show that < ru; v >=< u; −∇ · v >. Using (1:1) with the divergence theorem, Z Z < ru; v > = ru · v dx = − u r · v dx n n ZR R = u (−∇ · v) dx =< u; −∇ · v >; Rn and hence (r)∗ = −∇: completing the proof. 1.3 Integration by parts formula Let v be a vector field, and u be a scalar field. Then Z Z Z n v · ru dx = u v · n ds − u r · v dx; Ω ⊆ R and n unit normal vector: Ω @Ω Ω Proof. By integrating (1.1) on Ω we get, Z Z Z r · (u v) dx = u r · v dx + v · ru dx; Ω Ω Ω R R and by divergence theorem we know Ω r · (u v) dx = @Ω u v · n ds. Then Z Z Z u v · n ds = u r · v dx + v · ru dx; @Ω Ω Ω and hence Z Z Z v · ru dx = u v · n ds − u r · v dx: Ω @Ω Ω 4 1.4 Green's identities for local operators First Green's identity Let u; v are twice differentiable and n be a unit normal vector. Then Z Z u ∆v + ru · rv dx = u (rv · n)ds; (1.2) Ω @Ω Proof. Since r(uv) = u rv + vru and, r · (u rv) = u r · rv + ru · rv = u ∆v + ru · rv; By integrating on Ω we get, Z Z r · (u rv) dx = u ∆v + ru · rv dx; Ω Ω and by the divergence theorem, we obtain Z Z u rv · n ds = u ∆v + ru · rv dx: @Ω Ω Second Green's identity Let u; v are twice differentiable and n be a unit normal vector. Then Z Z u ∆v − v ∆u dx = (u rv − v ru) · n ds: Ω @Ω 5 Proof. Here we use (1.2) in the proof, Z Z Z Z u ∆v − v ∆u dx = u (rv · n) ds − ru · rv dx − v ∆u dx; Ω @Ω Ω Ω Z Z Z Z = u (rv · n) ds − ru · rv dx − v (ru · n) ds − rv · ru dx ; @Ω Ω @Ω Ω Z Z = u (rv · n) ds − v (ru · n) ds; @Ω @Ω Z = (u rv − vru) · n ds: @Ω 6 Chapter 2 Nonlocal Operators 2.1 Overview In this chapter and the following chapters, we deal with two different types of functions and two different types of nonlocal operators. A one-point scalar function u refers to a function u = u(x), with x 2 Rn, and a two-point scalar function u refers to a function defined for a pair of points u = u(x; y), with x; y 2 Rn. Similarly, a one-point vector field v refers to a vector field v = v(x), and a two-point vector field is of the form v = v(x; y). One-point operators map two-point functions to one-point functions, and two-point operators map one- point functions to two-point functions. In addition, symmetric and antisymmetric two-point scalar functions satisfy (x; y) = (y; x) and (x; y) = − (y; x), respectively, and similarly for symmetric and antisymmetric two-point vector fields. y−x n For example, let (x; y) = jjy−xjj , x 6= y; with x; y 2 R : Then is an antisymmetric two-point vector field. An example of a symmetric two-point scalar function is given by jjyjj+jjxjj n φ(x; y) = jjy−xjj , x 6= y; with x; y 2 R : Theorem 2.1. The following are equivalent: (i) the antisymmetry of a two-point function (x; y) = − (y; x); 8x; y 2 Rn. Z Z n (ii) (x; y)dydx = 0; 8 Ω ⊆ R . Ω Ω Z Z Z Z n (iii) (x; y)dydx + (x; y)dydx = 0; 8 Ω1; Ω2 ⊆ R . Ω1 Ω2 Ω2 Ω1 7 Proof. (i) ) (ii) Let (x; y) = − (y; x); 8 x; y 2 Rn then Z Z Z Z (x; y)dydx = − (y; x)dydx: Ω Ω Ω Ω By interchanging the variables (x $ y) in the second integral Z Z Z Z (x; y)dydx = − (x; y)dydx; Ω Ω Ω Ω Z Z and hence (x; y)dydx = 0. Ω Ω (ii) ) (i) Z Z n Assume (x; y)dydx = 0; 8 Ω ⊆ R . Then Ω Ω Z Z Z Z (x; y)dydx + (x; y)dydx = 0; Ω Ω Ω Ω By interchanging (x $ y) in the second integral, we obtain Z Z Z Z (x; y)dydx + (y; x)dydx = 0; Ω Ω Ω Ω Z Z n ( (x; y) + (y; x))dydx = 0; 8x; y 2 R ; Ω Ω which implies (x; y) + (y; x) = 0; and hence (x; y) = − (y; x): 8 x; y 2 Rn. (i) ) (iii) Assume (x; y) = − (y; x); 8 x; y 2 Rn.
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