A differential calculus for multifunctions
Differential calculus for multifunctions using Radstrom embedding theorem for convex sets
Banks, H. T.↗
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Nonlinear filtering for random signals in statistically unknown noise.
Natural and effective formulation of the filtering problem involved in satellite orbit determination, aircraft navigation, and missile tracking. The problem arises because the environment of the sensor keeps changing from time to time, and it is quite impractical and sometimes impossible to collect the statistical data of the noise incurred in the observation. Computable filtering equations are deduced. The idea of invariant imbedding along with stochastic differential calculus is used to derive differential equations for the optimal estimate.
Loo, J. T.↗
Two-Dimensional Mixed Problem of Thermoelasticity for a Semistrip
We study the stress state of a semistrip with conditions of smooth contact imposed on one of its lateral surfaces and coupling conditions given on the other surface under the action of normal loading and temperature on the end face. The original problem is reduced to a one-dimensional vector boundary-value problem by the method of integral transformations. By using the tools of matrix differential calculus and Green matrix function, we reduce the obtained problem to a singular integral equation for the derivative of displacements on the end face of the semistrip and solve this equation by the method of orthogonal polynomials. We perform the numerical analysis of fields of displacements and stresses inside the semistrip and establish the zones of tensile stresses on the lateral face of the semistrip and the conditions of their formation.
42 ENGINEERING↗
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative \(\mathbf {d}\) , Hodge star \(\star \) , and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator \(\overline{\mathbf {d}}\) and Hodge star operator \(\overline{\star }\) showing each converge spectrally to \(\mathbf {d}\) and \(\star \) . We demonstrate how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We introduce findings for the Laplace–Beltrami equations demonstrating our approach.
97 MATHEMATICS AND COMPUTING↗
A Sustainable Approach to Decision-Making in CCUS Systems
This study proposes a methodology for operational decision-making leading to the sustainability CO 2 -EOR systems. The methodology is a simple, yet comprehensive, integration of environmental and economic carbon capture, utilization and storage (CCUS) performance. Understanding the interplay between the environmental and economic conditions allows the EOR operator to manage the CO 2 -EOR flood so it remains within the limits of carbon balance neutrality and favorable economics under given prices and costs. The environmental performance and the main operational inputs (fluid production, CO 2 injection, and purchase requirements) were taken from a dynamic carbon Life Cycle Assessment (LCA) study previously performed for a CO 2 -EOR site in Cranfield, Mississippi. The LCA study accounts for greenhouse gases (GHG) emissions within a gate-to-grave CCUS system boundary, considering four different CO 2 injection strategies and four gas separation processes. The economic performance assessment was made through a marginalist analysis (Marginalist Production Theory), which determines important technical and economic relations through basic differential calculus derived from production, cost, and income functions. This method allows to define the operational economic optimum (optimum profit). The assessment considers four CO 2 price scenarios and uses World Bank estimated oil prices. In a first step of the proposed methodology, environmental limits of the operation are estimated. These limits are then compared with operative maximum productivity as well as with the economic optimum. According to the methodology, the EOR operation is sustainable as long as the environmental optimum is greater, or lasts longer, than the economic optimum along the project’s life. Results show that for this operation and assumed parameters, CO 2 -EOR could be classified as a sustainable activity depending on the operator’s strategic decision-making. In the case study, all injection strategies reach the environmental optimum after maximum productivity is reached. As a result, only two strategies (continuous gas injection and water alternating gas) find their economic optimum within the environmental limits, fulfilling the defined necessary and sufficient condition for CCUS sustainability.
02 PETROLEUM↗
Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation
We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
An endpoint sufficiency test
Endpoint sufficiency test based on differential calculus method, considering Bolza problem of continuous state and piecewise control variables
Vincent, T. L.↗
The geometric approach to sets of ordinary differential equations and Hamiltonian dynamics
The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.
Estabrook, F. B.↗
Differential geometric approaches to momentum-based formulations for fluids [Slides]
This SAND report documents CIS Late Start LDRD Project 22-0311, "Differential geometric approaches to momentum-based formulations for fluids". The project primarily developed geometric mechanics formulations for momentum-based descriptions of nonrelativistic fluids, utilizing a differential geometry/exterior calculus treatment of momentum and a space+time splitting. Specifically, the full suite of geometric mechanics formulations (variational/Lagrangian, Lie-Poisson Hamiltonian and Curl-Form Hamiltonian) were developed in terms of exterior calculus using vector-bundle valued differential forms. This was done for a fairly general version of semi-direct product theory sufficient to cover a wide range of both neutral and charged fluid models, including compressible Euler, magnetohydrodynamics and Euler-Maxwell. As a secondary goal, this project also explored the connection between geometric mechanics formulations and the more traditional Godunov form (a hyperbolic system of conservation laws). Unfortunately, this stage did not produce anything particularly interesting, due to unforeseen technical difficulties. There are two publications related to this work currently in preparation, and this work will be presented at SIAM CSE 23, at which the PI is organizing a mini-symposium on geometric mechanics formulations and structure-preserving discretizations for fluids. The logical next step is to utilize the exterior calculus based understanding of momentum coupled with geometric mechanics formulations to develop (novel) structure-preserving discretizations of momentum. This is the main subject of a successful FY23 CIS LDRD "Structure-preserving discretizations for momentum-based formulations of fluids".
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
On the continuation of the idea of heaviside in the operational calculus
Extension of classical methods of operational calculus with respect to theoretical assumptions and continuation of idea of Heaviside
CLASSICAL MECHANICS↗
Guidance applications of linear analysis
Differential equations of motion linearized about nominal calculus of variation solution to determine coefficients of guidance function
VARIATIONAL CALCULUS↗
Lectures on Lie Group Analysis: Solving Differential Equations Using Symmetries
These notes are meant to be a supplemental reference for the beginner Lie Group Analyst. It is assumed that the reader has a basic concept of the fundamentals of Lie Group Theory (LGT), e.g. has seen the derivation of the infinitesimal generator and understands the mathematical meaning behind invariance. An excellent reference is Albright et al., “Symmetry Analysis of Differential Equations: A Primer,”. The reader is urged to read at least the first three chapters of that document to be able to follow the outset of Chapter 2 of this document. The reader should also have a general understanding of calculus, ordinary differential equations, and partial differential equations.
97 MATHEMATICS AND COMPUTING↗
Varieties of operator manipulation
A change in MACSYMA syntax is proposed to accommodate the operator manipulators necessary to implement direct and indirect methods for the solution of differential equations, calculus of finite differences, and the fractional calculus, as well as their modern counterparts. To illustrate the benefits and convenience of this syntax extension, an example is given to show how MACSYMA's pattern-matching capability can be used to implement a particular set of operator identities which can then be used to obtain exact solutions to nonlinear differential equations.
Doohovskoy, A.↗
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications
Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology.
97 MATHEMATICS AND COMPUTING↗
Prym Differentials on Variable Tori
We consider multiplicative functions and Prym differentials on variable tori. We prove counterparts of the theorem on the total sum of residues for Prym differentials of any integer order on tori. As a consequence, reciprocity laws are proved. We construct elementary Prym differentials of all kinds with any integer order which holomorphically depend on the moduli of tori and characters. We derive an analogue of the Appell decomposition formula for functions with characters. We also study vector bundles of Prym differentials of any integer order over the product of Teichmueller spaces for a torus and a group of characters.
97 MATHEMATICS AND COMPUTING↗
Differential-Invariant Solutions
We construct differential-invariant solutions to automorphic systems and propose a method for classifying differential-invariant submodels.
97 MATHEMATICS AND COMPUTING↗
Adaptive Wavelet Decomposition of Matrix Flows
Adaptive algorithms for constructing spline-wavelet decompositions of matrix flows from a linear space of matrices over a normed field are presented. The algorithms suggested provides for an a priori prescribed estimate of the deviation of the basic flow from the initial one. Comparative bounds of the volumes of data in the basic flow for various irregularity characteristics of the initial flow are obtained in the cases of pseudo-equidistant and adaptive grids. Limit characteristics of the above-mentioned volumes are given in the cases where the initial flow is generated by differentiable functions.
97 MATHEMATICS AND COMPUTING↗