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https://digitalcommons.kettering.edu/mathematics_facultypubs
Recent documents in Mathematics Publicationsen-usFri, 19 Mar 2021 13:58:27 PDT3600Fostering student engagement through a real-world, collaborative project across disciplines and institutions
https://digitalcommons.kettering.edu/mathematics_facultypubs/105
https://digitalcommons.kettering.edu/mathematics_facultypubs/105Mon, 29 Jun 2020 09:29:50 PDT
Ample research has identified several features of a learning experience likely to enhance student learning, including collaboration, open-ended exploration, and problem-based learning in real-life scenarios. Missing is a model of how instructors might combine these elements into a single project that works flexibly across disciplines and institutions. This article fills this gap by offering such a model and reporting on its effectiveness in fostering student engagement. It describes a project that instructors at four colleges and universities in Flint, Michigan (USA) piloted during the height of the Flint water crisis. The project asked students to apply class content to the real-world problem unfolding around them, and offered students an opportunity to collaborate with peers. We collected qualitative and quantitative data on students’ reactions to the project, and found that the project succeeded in engaging students. We offer recommendations for how instructors can create similar projects in their own classrooms.
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Laura Mebert et al.The Role of Stretching in Slow Axonal Transport
https://digitalcommons.kettering.edu/mathematics_facultypubs/104
https://digitalcommons.kettering.edu/mathematics_facultypubs/104Sun, 07 Jun 2020 11:56:06 PDT
Axonal stretching is linked to rapid rates of axonal elongation. Yet the impact of stretching on elongation and slow axonal transport is unclear. Here, we develop a mathematical model of slow axonal transport that incorporates the rate of axonal elongation, protein half-life, protein density, adhesion strength, and axonal viscosity to quantify the effects of axonal stretching. We find that under conditions where the axon (or nerve) is free of a substrate and lengthens at rapid rates (>4 mm day−1), stretching can account for almost 50% of total anterograde axonal transport. These results suggest that it is possible to accelerate elongation and transport simultaneously by increasing either the axon's susceptibility to stretching or the forces that induce stretching. To our knowledge, this work is the first to incorporate the effects of stretching in a model of slow axonal transport. It has relevance to our understanding of neurite outgrowth during development and peripheral nerve regeneration after trauma, and hence to the development of treatments for spinal cord injury.
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Matthew O'Toole et al.Measurement of Subcellular Force Generation in Neurons
https://digitalcommons.kettering.edu/mathematics_facultypubs/103
https://digitalcommons.kettering.edu/mathematics_facultypubs/103Sun, 07 Jun 2020 11:55:59 PDT
Forces are important for neuronal outgrowth during the initial wiring of the nervous system and after trauma, yet subcellular force generation over the microtubule-rich region at the rear of the growth cone and along the axon has never, to our knowledge, been directly measured. Because previous studies have indicated microtubule polymerization and the microtubule-associated proteins Kinesin-1 and dynein all generate forces that push microtubules forward, a major question is whether the net forces in these regions are contractile or expansive. A challenge in addressing this is that measuring local subcellular force generation is difficult. Here we develop an analytical mathematical model that describes the relationship between unequal subcellular forces arranged in series within the neuron and the net overall tension measured externally. Using force-calibrated towing needles to measure and apply forces, in combination with docked mitochondria to monitor subcellular strain, we then directly measure force generation over the rear of the growth cone and along the axon of chick sensory neurons. We find the rear of the growth cone generates 2.0 nN of contractile force, the axon generates 0.6 nN of contractile force, and that the net overall tension generated by the neuron is 1.3 nN. This work suggests that the forward bulk flow of the cytoskeletal framework that occurs during axonal elongation and growth-cone pauses arises because strong contractile forces in the rear of the growth cone pull material forward.
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Matthew O'Toole et al.Angled derivative approximation of the hyperbolic heat conduction equations
https://digitalcommons.kettering.edu/mathematics_facultypubs/102
https://digitalcommons.kettering.edu/mathematics_facultypubs/102Sun, 19 Apr 2020 10:33:49 PDT
Numerical methods based upon angled derivative approximation are presented for a linear first-order system of hyperbolic partial differential equations (PDEs): the hyperbolic heat conduction equations. These equations model the flow of heat in circumstances where the speed of thermal propagation is finite as opposed to the infinite wave speed inherent in the diffusion equation. A basic angled derivative scheme is first developed which is second-order accurate. From this, an enhanced angled derivative scheme is then developed which is fourth-order accurate for Courant numbers of one-half and one. Both methods are explicit three-level schemes, which are conditionally stable. Careful treatment of initial and boundary conditions is provided which preserves overall order of accuracy and stability and suppresses any deleterious effects of spurious modes. After establishing a necessary and sufficient stability condition of Courant number less than or equal to one for both schemes, their dissipative and dispersive properties are investigated. A numerical example concerning the propagation of a thermal pulse train concludes this investigation.
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Brian J. McCartin et al.Higher Order A-Stable Schemes for the Wave Equation Using a Successive Convolution Approach
https://digitalcommons.kettering.edu/mathematics_facultypubs/101
https://digitalcommons.kettering.edu/mathematics_facultypubs/101Sun, 19 Apr 2020 10:33:43 PDT
In several recent works, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL$^T$) formulation combined with alternating direction implicit (ADI) schemes. Because our method is based on an integral solution of the ADI splitting of the MOL$^T$ formulation, we are able to easily embed non-Cartesian boundaries and include point sources with exact spatial resolution. Further, we developed an efficient $O(N)$ convolution algorithm for rapid evaluation of the solution, which makes our method competitive with explicit finite difference (e.g., finite difference time domain) solvers, in terms of both accuracy and time to solution, even for Courant numbers slightly larger than 1. We have demonstrated the utility of this method by applying it to a range of problems with complex geometry, including cavities with cusps. In this work, we present several important modifications to our recently developed wave solver. We obtain a family of wave solvers which are unconditionally stable, accurate of order $2P$, and require $O(P^d N)$ operations per time step, where $N$ is the number of spatial points and $d$ the number of spatial dimensions. We obtain these schemes by including higher derivatives of the solution, rather than increasing the number of time levels. The novel aspect of our approach is that the higher derivatives are constructed using successive applications of the convolution operator. We develop these schemes in one spatial dimension, and then extend the results to higher dimensions, by reformulating the ADI scheme to include recursive convolution. Thus, we retain a fast, unconditionally stable scheme, which does not suffer from the large dispersion errors characteristic to the ADI method. We demonstrate the utility of the method by applying it to a host of wave propagation problems. This method holds great promise for developing higher order, parallelizable algorithms for solving hyperbolic PDEs and can also be extended to parabolic PDEs.
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Matthew F. Causley et al.An Infinite-Horizon Stochastic Optimal Control Model for Online Seller Behavior
https://digitalcommons.kettering.edu/mathematics_facultypubs/100
https://digitalcommons.kettering.edu/mathematics_facultypubs/100Sun, 19 Apr 2020 10:33:36 PDT
In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (online store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller's reputation. We find that for each seller there is an optimal reputation, i.e. the reputation at which the seller should switch her state in order to maximize her total profit. Relying on techniques of stochastic optimal control, we design a stochastic behavioral model for an online seller. This model incorporates the dynamics of resource allocation and reputation. We optimize the design model by using a stochastic advertising model put forth by Sethi (Optimal Control Applications and Methods, vol. 4, no. 2, 1983, pp. 179-184) and used effectively in Raman's more recent work (Automatica, vol. 42, no. 8, 2006, pp. 1357-1362). This model of reputation is combined with the effect of online reputation on sales price empirically verified by by Mink and Seifert (Group Decision and Negotiation International Conference, June 2006, pp. 253-255), and we derive the resulting Hamilton-Jacobi-Bellman (HJB) differential equation. The solution of this HJB equation relates optimal wealth level to a seller's reputation.
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Milan Bradonij' et al.Stochastic Optimal Control for Online Seller under Reputational Mechanisms
https://digitalcommons.kettering.edu/mathematics_facultypubs/99
https://digitalcommons.kettering.edu/mathematics_facultypubs/99Sun, 19 Apr 2020 10:33:29 PDT
In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller’s reputation. We show that for each seller there is an optimal reputation, i.e., the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which incorporates the dynamics of resource allocation and reputation. The design of the model is optimized by using a stochastic advertising model from [1] and used effectively in the Stochastic Optimal Control of Advertising [2]. This model of reputation is combined with the effect of online reputation on sales price empirically verified in [3]. We derive the Hamilton-Jacobi-Bellman (HJB) differential equation, whose solution relates optimal wealth level to a seller’s reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of the optimal seller behavior. Coincidentally, the reduced model has a closed form analytical solution that we construct.
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Milan Bradonij' et al.Efficient high-order methods for solving fractional differential equations of order α ∈ (0, 1) using fast convolution and applications in wave propagation
https://digitalcommons.kettering.edu/mathematics_facultypubs/98
https://digitalcommons.kettering.edu/mathematics_facultypubs/98Sun, 19 Apr 2020 10:33:23 PDT
In this work we develop a means to rapidly and accurately compute the Caputo fractional derivative of a function, using fast convolution. The key element to this approach is the compression of the fractional kernel into a sum of M decaying exponentials, where M is minimal. Specifically, after N time steps we find M= O (log N) leading to a scheme with O (N log N) complexity. We illustrate our method by solving the fractional differential equation representing the Kelvin-Voigt model of viscoelasticity, and the partial differential equations that model the propagation of electromagnetic pulses in the Cole-Cole model of induced dielectric polarization.
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Matthew F. Causley et al.Method of lines transpose: an efficient A-stable solver for wave propagation
https://digitalcommons.kettering.edu/mathematics_facultypubs/97
https://digitalcommons.kettering.edu/mathematics_facultypubs/97Sun, 19 Apr 2020 10:33:16 PDT
Building upon recent results obtained in [7,8,9], we describe an efficient second order, A-stable scheme for solving the wave equation, based on the method of lines transpose (MOLT), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In [7], A-stable schemes of high order were derived, and in [9] a high order, fast O(N) spatial solver was derived, which is matrix-free and is based on dimensional-splitting. In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOLT formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides.
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Matthew Causley et al.Method of Lines Transpose: High Order L-Stable {O}(N) Schemes for Parabolic Equations Using Successive Convolution
https://digitalcommons.kettering.edu/mathematics_facultypubs/96
https://digitalcommons.kettering.edu/mathematics_facultypubs/96Sun, 19 Apr 2020 10:33:10 PDT
We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a one-dimensional heat equation solver that uses fast $\mathcal O(N)$ convolution. This fundamental solver has arbitrary order of accuracy in space and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multidimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite--Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen--Cahn, and the FitzHugh--Nagumo system of equations in one and two dimensions.
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Matthew F. Causley et al.Method of Lines Transpose: An Efficient Unconditionally Stable Solver for Wave Propagation
https://digitalcommons.kettering.edu/mathematics_facultypubs/95
https://digitalcommons.kettering.edu/mathematics_facultypubs/95Sun, 19 Apr 2020 10:33:03 PDT
Building upon recent results obtained in Causley and Christlieb (SIAM J Numer Anal 52(1):220–235, 2014), Causley et al. (Math Comput 83(290):2763–2786, 2014, Method of lines transpose: high order L-stable O(N) schemes for parabolic equations using successive convolution, 2015), we describe an efficient second-order, unconditionally stable scheme for solving the wave equation, based on the method of lines transpose (MOLTT), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In Causley and Christlieb (SIAM J Numer Anal 52(1):220–235, 2014), unconditionally stable schemes of high order were derived, and in Causley et al. (Method of lines transpose: high order L-stable O(N) schemes for parabolic equations using successive convolution, 2015) a high order, fast O(N)O(N) spatial solver was derived, which is matrix-free and is based on dimensional-splitting. In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOLTT formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides.
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Matthew Causley et al.A particle-in-cell method for the simulation of plasmas based on an unconditionally stable field solver
https://digitalcommons.kettering.edu/mathematics_facultypubs/94
https://digitalcommons.kettering.edu/mathematics_facultypubs/94Sun, 19 Apr 2020 10:32:55 PDT
We propose a new particle-in-cell (PIC) method for the simulation of plasmas based on a recently developed, unconditionally stable solver for the wave equation. This method is not subject to a CFL restriction, limiting the ratio of the time step size to the spatial step size, typical of explicit methods, while maintaining computational cost and code complexity comparable to such explicit schemes. We describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases, and present the results of several standard test problems, showing good agreement with theory with time step sizes much larger than allowed by typical CFL restrictions.
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Eric M. Wolf et al.Sequences which converge to e: New insights from an old formula
https://digitalcommons.kettering.edu/mathematics_facultypubs/93
https://digitalcommons.kettering.edu/mathematics_facultypubs/93Sun, 19 Apr 2020 10:32:49 PDT
One of the most fundamental results in calculus was the discovery of the mathematical constant e = 2.718... by Jacob Bernoulli. Remarkably, new definitions of e are still being discovered, in part due to renewed interest at the advent of modern computing and the quest for more digits. In this work we review recent discoveries of sequences which tend to e, and propose a systematic approach for producing such sequences. In doing so, we establish several classes of sequences, and their generalizations. Our methods use only basic tools of calculus and numerical analysis, such as series expansions and Padé approximants. Numerical results demonstrate that our new sequences rapidly converge to e.
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Matthew F. Causley et al.Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models
https://digitalcommons.kettering.edu/mathematics_facultypubs/92
https://digitalcommons.kettering.edu/mathematics_facultypubs/92Sun, 19 Apr 2020 10:32:42 PDT
In this work, we develop an $\mathcal{O}(N)$ implicit real space method in 1D and 2D for the Cahn--Hilliard (CH) and vector Cahn--Hilliard (VCH) equations, based on the method of lines transpose (MOL$^{T}$) formulation. This formulation results in a semidiscrete time stepping algorithm, which we prove is gradient stable in the $H^{-1}$ norm. The spatial discretization follows from dimensional splitting and an $\mathcal{O}(N)$ matrix-free solver, which applies fast convolution to the modified Helmholtz equation. We propose a novel factorization technique, in which fourth-order spatial derivatives are incorporated into the solver. The splitting error is included in the nonlinear fixed point iteration, resulting in a high-order, logically Cartesian (line-by-line) update. Our method is fast but not restricted to periodic boundaries like the fast Fourier transform (FFT). The basic solver is implemented using the backward Euler formulation, and we extend this to both backward difference formula (BDF) stencils, singly diagonal implicit Runge--Kutta (SDIRK), and spectral deferred correction (SDC) frameworks to achieve high orders of temporal accuracy. We demonstrate with numerical results that the CH and VCH equations maintain gradient stability in one and two spatial dimensions. We also explore time-adaptivity, so that meta-stable states and ripening events can be simulated both quickly and efficiently.
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Matthew Causley et al.On the convergence of spectral deferred correction methods
https://digitalcommons.kettering.edu/mathematics_facultypubs/91
https://digitalcommons.kettering.edu/mathematics_facultypubs/91Sun, 19 Apr 2020 10:32:35 PDT
In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this high-order ordinary differential equation (ODE) solver is typically described wherein a low-order approximation (such as forward or backward Euler) is lifted to higher order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard integral equation solver. In doing so, our chief finding is that it is not the low-order solver that picks up the order of accuracy with each correction, but it is the underlying quadrature rule of the right hand side function that is solely responsible for picking up additional orders of accuracy. Our proofs point to a total of three sources of errors that SDC methods carry: the error at the current time point, the error from the previous iterate, and the numerical integration error that comes from the total number of quadrature nodes used for integration. The second of these two sources of errors is what separates SDC methods from Picard integral equation methods; our findings indicate that as long as difference between the current and previous iterate always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying "solver" is inconsistent the underlying ODE. From this vantage, we solidify the prospects of extending spectral deferred correction methods to a larger class of solvers to which we present some examples.
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Matthew F. Causley et al.Propagation of Smoothness for Edge-degenerate Wave Equations
https://digitalcommons.kettering.edu/mathematics_facultypubs/90
https://digitalcommons.kettering.edu/mathematics_facultypubs/90Mon, 13 Apr 2020 15:04:08 PDT
A propagation result for an edge-degenerate wave equation in 1+1 dimensions is proved. The proposed method generalizes to other edge-degenerate wave equations.
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Ruben G. Hayrapetyan et al.Isometric properties of the Hankel transformation in weighted Sobolev spaces
https://digitalcommons.kettering.edu/mathematics_facultypubs/89
https://digitalcommons.kettering.edu/mathematics_facultypubs/89Mon, 13 Apr 2020 15:04:02 PDT
It is shown that the Hankel transformation H v acts in a class of weighted Sobolev spaces. Especially, the isometric mapping property of H v which holds on L ² is extended to spaces of arbitrary Sobolev order. The novelty in the approach consists in using techniques developed by B.-W. Schulze and others to treat the half-line as a manifold with a conical singularity at r = 0. This is achieved by pointing out a connection between the Hankel transformation and the Mellin transformation. The procedure proposed leads at the same time to a short proof of the Hankel inversion formula. An application to the existence and higher regularity of solutions, including their asymptotics, to the 1+1 dimensional edge-degenerate wave equation is given.
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Ruben G. Hayrapetyan et al.Kinetics approach to modeling of polymer additive degradation in lubricants
https://digitalcommons.kettering.edu/mathematics_facultypubs/88
https://digitalcommons.kettering.edu/mathematics_facultypubs/88Mon, 13 Apr 2020 15:03:55 PDT
A kinetics problem for a degrading polymer additive dissolved in a base stock is studied. The polymer degradation may be caused by the combination of such lubricant flow parameters as pressure, elongational strain rate, and temperature as well as lubricant viscosity and the polymer characteristics (dissociation energy, bead radius, bond length, etc.). A fundamental approach to the problem of modeling mechanically induced polymer degradation is proposed. The polymer degradation is modeled on the basis of a kinetic equation for the density of the statistical distribution of polymer molecules as a function of their molecular weight. The integrodifferential kinetic equation for polymer degradation is solved numerically. The effects of pressure, elongational strain rate, temperature, and lubricant viscosity on the process of lubricant degradation are considered. The increase of pressure promotes fast degradation while the increase of temperature delays degradation. A comparison of a numerically calculated molecular weight distribution with an experimental one obtained in bench tests showed that they are in excellent agreement with each other.
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Ilya Kudish et al.Isothermal EHL problem for chemically degrading lubricant
https://digitalcommons.kettering.edu/mathematics_facultypubs/87
https://digitalcommons.kettering.edu/mathematics_facultypubs/87Mon, 13 Apr 2020 15:03:49 PDT
A new formulation for an elastohydrodynamic problem with degrading lubricant is proposed. The formulation takes into account stress-induced degradation of polymer additive to lubricant on the basis of the kinetic equation for polymer degradation. Degradation of polymer additive results in an irreversible viscosity loss, which, in turn, leads to a reduced lubrication film thickness and change in other contact parameters. The problem is solved numerically. A solution of the kinetics equation is compared with some experimental polymer degradation data. The calculated value of the viscosity loss in a lubricated contact is similar to an earlier obtained one in experiments.
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Ilya I. Kudish et al.Modeling of Kinetics of Stress-Induced Degradation of Polymer Additives in Lubricants and Viscosity Loss
https://digitalcommons.kettering.edu/mathematics_facultypubs/86
https://digitalcommons.kettering.edu/mathematics_facultypubs/86Mon, 13 Apr 2020 15:03:42 PDT
A fundamental approach to the problem of modeling mechanically induced polymer degradation is proposed. The polymer degradation is modeled by a kinetic equation for the density of the statistical distribution of linear polymer molecules as a function of their molecular weight. The integrodifferential kinetic equation is solved numerically. A comparison of numerically calculated molecular weight distributions and lubricant viscosity loss caused by polymer degradation with experimental ones obtained in bench tests showed that they are in excellent agreement. The effects of pressure, shear, temperature, and lubricant viscosity on lubricant degradation are considered. The increase of pressure promotes fast degradation while the increase of temperature depending on other parameters may delay or promote degradation. In some cases, the density of the molecular weight distribution function maintained its initial single-modal shape and in other cases it changed with time from a single-modal shape to a multi-modal one.
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Ilya I. Kudish et al.