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The Multi-Index Notation
Published in M. W. Wong, Partial Differential Equations, 2022
We begin with the standard multi-index notation in the modern theory of partial differential equations. Let x=(x1,x2,…,xn) and y=(y1,y2,…,yn) be points in the Euclidean space ℝn. Then the dot product x·y of x and y is defined by x·y=∑j=1nxjyj
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
In this book we shall adopt the multi-index notation for the partial derivatives of higher order of the following form: ADαf=∂|α|f∂xα $$ A\boldsymbol{D}^{\boldsymbol{\alpha }}\boldsymbol{f}= \frac{\partial ^{|\boldsymbol{\alpha }|}\boldsymbol{f}}{\partial \boldsymbol{x}^{\boldsymbol{\alpha }}} $$
Hilbert spaces
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Exercise 9.4. Assume xj ≥ 0 and aj > 0 for 1 ≤ j ≤ n. Find the maximum of ∏j=1nxjaj given that ∑xj=1. Comment: In two dimensions, the maximum of xa yb on x + y = 1 is aabb(a+b)a+b. The answer is αα|α||α| in multi-index notation, which we do not discuss. See page 4 of [T], page 227 of [F1], page 64 of [D1], or many other sources for multi-index notation.
Constrained variational problems governed by second-order Lagrangians
Published in Applicable Analysis, 2020
According to Lagrange theory, an extreme point of (P2) is found among the extreme points of (P2). In order to formulate the necessary conditions of optimality, we shall introduce the Saunders's multi-index notation [12]. A multi-index is an m-tuple I of natural numbers. The components of I are denoted , where α is an ordinary index, . The multi-index is defined by for . The addition and the substraction of the multi-indexes are defined componentwise (although the result of a substraction might not be a multi-index): . The length of a multi-index is , and its factorial is . The number of distinct indices represented by , , is Particular cases.
Multidimensional inverse Cauchy problems for evolution equations
Published in Inverse Problems in Science and Engineering, 2020
Mukhtar Karazym, Tohru Ozawa, Durvudkhan Suragan
Let us consider the higher order linear partial differential equation with the first-order partial time derivative where in multi-index notation with and . We assume that the coefficients are sufficiently smooth and bounded if necessary. Also we assume that the coefficients of the highest derivatives are non-zero everywhere in . The solution of (15) with the initial condition is given by where is the fundamental solution, see [15, Section 9.6.3-1]. We assume that the solution of (15)–(16) is unique and the fundamental solution satisfies Note that, the fundamental solution is a solution of We also assume that the operator satisfies Green's second identity, that is, for all and . We solve the following inverse problem of finding a unique pair and where with and are sufficiently smooth such that for all which implies .
Model reduction of parabolic PDEs using multivariate splines
Published in International Journal of Control, 2019
H. J. Tol, C. C. de Visser, M. Kotsonis
with y(x,t) the state variable, x ∈ Ω the spatial coordinate, a measured output and a controlled output which is used to define the control objective later in this section. The vector function g(x) = [g1(x),… , gm(x)], describes how the inputs from m linear actuators are distributed in the domain, describes how the inputs are distributed over the boundary and φi(x) is determined by the desired performance specifications in the domain Ω. The operator L is defined as a linear partial differential operator with derivatives up to order k ≥ 1 with spatially varying coefficients where we have used the well-known multi-index notation for the spatial derivative for a given multi-index (α1, α2,… , αn) of order |α| = α1 + α2 + ⋅⋅⋅ + αn and the operators are defined as partial differential operators with constant coefficients. Common boundary conditions are Dirichlet (), Neumann () and Robin boundary conditions (). In this study, feedback stabilisation of (1a) is considered where the PDE describes the error between the unsteady response and the equilibrium profile, e.g. the error between the unsteady temperature and the equilibrium profile of the temperature. It is assumed that point measurements from K boundary or in-domain sensors are used for feedback.