Class Of Ordinary Differential Equations Research Articles

An existence and uniqueness result about algebras of Schwartz distributions

We prove that there exists essentially one minimal differential algebra of distributions A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal A$$\\end{document}, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l’impossibilité de la multiplication des distributions, 1954], and such that Cp∞⊆A⊆D′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal C_p^{\\infty } \\subseteq \\mathcal A\\subseteq \\mathcal D' $$\\end{document} (where Cp∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal C_p^{\\infty }$$\\end{document} is the set of piecewise smooth functions and D′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D'$$\\end{document} is the set of Schwartz distributions over R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R$$\\end{document}). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal A$$\\end{document}.

Statistical determinism in non-Lipschitz dynamical systems

Abstract We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $\nu $ : the regularized dynamics is globally defined for each $\nu> 0$ , and the original singular system is recovered in the limit of vanishing $\nu $ . We prove that this limit yields a unique statistical solution independent of regularization when the deterministic system possesses a chaotic attractor having a physical measure with the convergence to equilibrium property. In this case, solutions become spontaneously stochastic after passing through the singularity: they are selected randomly with an intrinsic probability distribution.

A methodology using a non-increasing sequence applied in the solution of heat transfer problems

This research aims to propose a numerical methodology that employs a non-increasing sequence to approximate the solution for a large class of ordinary differential equations like the ones described in complex non-linear heat transfer through porous fins. It also aims to compare this proposed methodology with an earlier one using a non-decreasing sequence of elements and to provide an upper-bound estimation for the error. The comparison between the non-increasing and non-decreasing sequences showed excellent agreement when applied to an example of convection and radiation in porous fins.

Stochastic Delay Differential Equations: A Comprehensive Approach for Understanding Biosystems with Application to Disease Modelling

Mathematical models have been of great importance in various fields, especially for understanding the dynamical behaviour of biosystems. Several models, based on classical ordinary differential equations, delay differential equations, and stochastic processes are commonly employed to gain insights into these systems. However, there is potential to extend such models further by combining the features from the classical approaches. This work investigates stochastic delay differential equations (SDDEs)-based models to understand the behaviour of biosystems. Numerical techniques for solving these models that demonstrate a more robust representation of real-life scenarios are presented. Additionally, quantitative roles of delay and noise to gain a deeper understanding of their influence on the system’s overall behaviour are analysed. Subsequently, numerical simulations that illustrate the model’s robustness are provided and the results suggest that SDDEs provide a more comprehensive representation of many biological systems, effectively accounting for the uncertainties that arise in real-life situations.

Near-wall model for compressible turbulent boundary layers based on an inverse velocity transformation

In this work, a near-wall model, which couples the inverse of a recently developed compressible velocity transformation (Griffin et al., Proc. Natl Acad. Sci., vol. 118, 2021, p. 34) and an algebraic temperature–velocity relation, is developed for high-speed turbulent boundary layers. As input, the model requires the mean flow state at one wall-normal height in the inner layer of the boundary layer and at the boundary-layer edge. As output, the model can predict mean temperature and velocity profiles across the entire inner layer, as well as the wall shear stress and heat flux. The model is tested in an a priori sense using a wide database of direct numerical simulation high-Mach-number turbulent channel flows, pipe flows and boundary layers (48 cases, with edge Mach numbers in the range 0.77–11, and semi-local friction Reynolds numbers in the range 170–5700). The present model is significantly more accurate than the classical ordinary differential equation (ODE) model for all cases tested. The model is deployed as a wall model for large-eddy simulations in channel flows with bulk Mach numbers in the range 0.7–4 and friction Reynolds numbers in the range 320–1800. When compared to the classical framework, in the a posteriori sense, the present method greatly improves the predicted heat flux, wall stress, and temperature and velocity profiles, especially in cases with strong heat transfer. In addition, the present model solves one ODE instead of two, and has a computational cost and implementation complexity similar to that of the commonly used ODE model.

Long time decay analysis of complex-valued fractional abstract evolution equations with delay

The asymptotic stability and long time decay rates of solutions to linear Caputo time-fractional ordinary differential equations are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of time-fractional ordinary differential equations decay only polynomially, leading to the so-called Mittag-Leffler stability, which were already extended to linear fractional delay differential equations (FDDEs) with real constants by Čermák et al. (2017) [17]. This paper is devoted to the asymptotical stability and long time decay rates of the solutions for a class of fractional delay abstract evolution equations with complex-valued coefficients. By applying the spectral decomposition method, we reformulate the fractional delay abstract evolution equations to an infinite-dimensional FDDEs. Through the root locus technique, we develop a necessary and sufficient stability condition in a coefficient criterion for FDDEs. At the same time, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long time decay rates of the solutions for both linear FDDEs and fractional delay abstract evolution equations with complex-valued coefficients are O(t−α). Finally, several application models including both time and time-space fractional Schrödinger equations are presented to illustrate the effectiveness of our proposed criteria.

Model predictive control of nonlinear processes using neural ordinary differential equation models

Neural Ordinary Differential Equation (NODE) is a recently proposed family of deep learning models that can perform a continuous approximation of a linear/nonlinear dynamic system using time-series data by integrating the neural network model with classical ordinary differential equation solvers. Modeling nonlinear dynamic processes using data has historically been a critical challenge in the field of chemical engineering research. With the development of computer science and neural network technology, recurrent neural networks (RNN) have become a popular black-box approach to accomplish this task and have been utilized to design model predictive control (MPC) systems. However, as a discrete-time approximation model, RNN requires strictly uniform step sequence data to operate, which makes it less robust to an irregular sampling scenario, such as missing data points during operation due to sensor failure or other types of random errors. This paper aims to develop a NODE model and construct an MPC based on this novel continuous-time neural network model. In this context, closed-loop stability is established and robustness to noise is addressed using a variety of data analysis techniques. An example of a chemical process is utilized to evaluate the performance of NODE-based MPC. Furthermore, the performance of the NODE-based MPC under Gaussian and non-Gaussian noise is investigated, and the subsampling method is found to be effective in building models for MPC that are suitable for handling the presence of non-Gaussian noise in the data.

On the scaling behaviour of an integrable spin chain with [formula omitted] symmetry

The subject matter of this work is a 1D quantum spin-12 chain associated with the inhomogeneous six-vertex model possessing an additional Zr symmetry. The model is studied in a certain parametric domain, where it is critical. Within the ODE/IQFT approach, a class of ordinary differential equations and a quantization condition are proposed which describe the scaling limit of the system. Some remarkable features of the CFT underlying the critical behaviour are observed. Among them is an infinite degeneracy of the conformal primary states and the presence of a continuous component in the spectrum in the case of even r.

Stability and asymptotics for fractional delay diffusion‐wave equations

The asymptotic stability and long‐time decay rates of solutions to linear Caputo time‐fractional ordinary differential equations (FODEs) with order both and are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of FODEs decay only polynomially, leading to the so‐called Mittag–Leffler stability, which was already extended to linear fractional delay differential equations (FDDEs) with order . However, the study on the long‐time decay rates of the solutions for FDDEs with order is scarce. Hence, this paper is devoted to the asymptotical stability and long‐time decay rates for a class of FDDEs with , that is, fractional delay diffusion‐wave equations. We reformulate the fractional delay diffusion‐wave equations to infinite‐dimensional FDDEs by the eigenvalue decomposition method. Through a detailed analysis for the characteristic equations, a novel necessary and sufficient stability condition is established in a coefficient criterion. Moreover, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long‐time decay rates of the solutions for both linear FDDEs and fractional delay diffusion‐wave equations are . Finally, an example of an application model with feedback control is provided to show the effectiveness of our theoretical results.

Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film

Abstract Numerous flow and heat transfer studies have relied on the construction of similarity transformations which map the nonlinear partial differential equations (PDEs) describing the flow and heat transfer, to ordinary differential equations (ODEs). For these reduced equations, one finds multiple analytic and approximate solution procedures as compared to the flow PDEs. Here, we aim at constructing multiple classes of similarity transformations that are different from those already existing in the literature. We adopt the Lie symmetry method to derive these new similarity transformations which reveal new classes of ODEs corresponding to flow equations when applied to them. With these multiple classes of similarity transformations, one finds multiple reductions in the flow PDEs to ODEs. On solving these ODEs analytically or numerically, we obtain different kinds of flow and heat transfer patterns that help in determining optimized solutions in accordance with the physical requirements of a problem. For the said purpose, we derive Lie point symmetries for the magnetohydrodynamic Casson fluid flow and heat transfer in a thin film on an unsteady stretching sheet with viscous dissipation. Linear combinations of these Lie symmetries that are again the Lie symmetries of the flow model are employed here to construct new similarity transformations. We derive multiple Lie similarity transformations through the proposed procedure which lead us to more than one class of reduced ODEs obtained by applying the deduced transformations. We analyze the flow and heat transfer by deriving analytic solutions for the obtained classes of systems of ODEs using the homotopy analysis method. Magnetic parameters and viscous dissipation influences on the flow and heat transports are investigated and presented in graphical and tabulated formats.

What explains the poor contraction of the viral load during paediatric HIV infection?

An acute HIV infection in young children differs markedly from that in adults: Children have higher viral loads (VL), and a poor contraction to a setpoint VL that is not much lower than the peak VL. As a result, children progress faster towards AIDS in the absence of treatment. We used a classical ordinary differential equation model for viral infection dynamics to study why children have a lower viral contraction ratio than adults. We performed parameter sweeps to identify factors explaining the observed difference between children and adults. We grouped parameters associated with the host, the infection, or the immune response. Based on paediatric data available from datasets within the EPIICAL project (https://www.epiical.org/), we refuted that viral replication rates differ between young children and adults, and therefore these cannot be responsible for the low VL contraction ratios seen in children. The major differences in lowering VL contraction ratio resulted from sweeping the parameters linked to the immune response. Thus, we postulate that an ”ineffective” (late and/or weak) immune response is the most parsimonious explanation for the higher setpoint VL in young children, and hence the reason for their fast disease progression.

Applying a Power Transformation to the Orbit of the 2nd Painleve Equation and Solving Differential Equations with Polynomial Right-hand Sides Via the 2nd Painleve Transcendent and in Polynomials

The 2nd Painleve equation is considered as a representative of the second-order class of ordinary differential equations (ODEs) with polynomial right-hand sides, as well as of the more general second-order class of equations with fractional polynomial right-hand sides. The second Painlevé equation with three terms on the right side has an orbit in the class of fractional polynomial equations with respect to the pseudogroup of the 36th order, and in the absence of the 3rd term – the 60th order. This paper presents a power transformation with an arbitrary parameter that preserves the polynomial or fractional polynomial form of the equations. This power-law transformation is applied to the orbital equations of the 2nd Painlevé equation with three and two terms on the right-hand sides of the equations. Pseudogroups of transformations induced by the above-mentioned pseudogroups of the 36th and 60th orders are constructed. All equations with one-constant arbitrariness corresponding to the vertices of the graphs of induced pseudogroups are found. General solutions of all found equations are obtained through the 2nd Painlevé transcendental or in polynomials. A theorem is presented that allows, using the scaling operation, to find general solutions to all the above equations with arbitrary coefficients.

Analysis of fractional global differential equations with power law

<abstract><p>We have considered a special class of ordinary differential equations in which the differential operators are those of the Caputo fractional global derivative. These equations are generalizations of the well-known differential equations with the Caputo fractional derivative. Due to the various possible applications of these equations to model real-world problems we have first introduced some new inequalities that will be used in all fields of science, technology and engineering where these equations could be applied. We used Nagumo's principles to establish the existence and uniqueness of the solution for this class of equations with additional conditions. We have applied the midpoint principle to obtain a numerical scheme that will be used to solve these equations numerically. Some illustrative examples are presented with excellent results.</p></abstract>

Classes of Ordinary Differential Equations of Length Biased Exponential Distribution and their Solutions

The purpose of the work is to generate the ordinary differential equations and their solutions for the probability density function, quantile rate function, survival rate function, inverse survival rate function, haz-ard rate function and reversed hazard rate function of the Length Biased Exponential Distribution. The ordinary differential equations and their so-lutions are obtained using the Math of differentiation and integration as a tool together with their boundary conditions. The boundaries and para-meters that portrayed the distribution unavoidably decide the nature, presence, uniqueness, arrangement and the various possible solutions of these ordinary differential equations are better approaches to understand these characteristics. The work will be helpful to analyze the lifetime growth or risk and is of great significance in the field of ecological studies. The method can be very useful for the other probability distributions and can serve as a substitute for the approximation study.

Combination of anti-PD-L1 and radiotherapy in hepatocellular carcinoma: a mathematical model with uncertain parameters

Blockade of programmed death-ligand-1 (PD-L1) as a new method of immunotherapy for cancers has shown limited efficacy in hepatocellular carcinoma (HCC). The combination of anti-PD-L1 and radiotherapy (RT) enhances the antitumor effect in HCC cancer. The efficacy and interactions of these treatments can be addressed by a mathematical model. We developed a mathematical model using a set of ordinary differential equations (ODEs). The variables include cancer cells, cytotoxic T lymphocytes (CTLs), programmed cell death-1 (PD-1), PD-L1, anti-PD-L1, and ionizing radiation. The model is parameterized with imprecise data set of murine HCC model and the effect of parametric uncertainty is assessed by the fuzzy theorem. The global sensitivity analysis (GSA) is performed to assess model robustness against perturbation in parameters and to identify the most influential parameters on the dynamics of cells and proteins. In silico predictions are consistent with experimental data. The model simulation shows that anti-PD-L1 and RT have a synergistic effect. In silico assessment of treatments’ efficacy in the fuzzy setting of parameters revealed that anti-PD-L1 therapy, RT, and combination treatment caused the uncertainty band of tumor cells to lead to lower populations. This model as a validated rigorous simulation framework can be used to deepen our understanding of tumor and immune cell interactions and helps clinicians to investigate the efficacy of different time schedules of anti-PD-L1, RT, and combination therapy. The fuzzy theorem in conjunction with the classical ODE model that is parameterized by imprecise data was used to predict reliable outcomes of treatment.

Fractal analysis of degenerate spiral trajectories of a class of ordinary differential equations

In this paper we initiate the study of the Minkowski dimension, also called the box dimension, of degenerate spiral trajectories of a class of ordinary differential equations. A class of singularities of focus type with two zero eigenvalues (nilpotent or more degenerate) has been studied. We find the box dimension of a polynomial degenerate focus of type (n,n) by exploiting the well-known fractal results for α-power spirals. In the general (m,n) case, we formulate a conjecture about the box dimension of a degenerate focus using numerical experiments. Further, we reduce the fractal analysis of planar nilpotent contact points to the study of the box dimension of a slow-fast spiral generated by their “entry-exit” function. There exists a bijective correspondence between the box dimension of the slow-fast spirals and the codimension of contact points. We also construct a three-dimensional vector field that contains a degenerate spiral, called an elliptical power spiral, as a trajectory.

Convergence of the Fourth Order Variable Step Size Super Class of Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

In many fields of study such as science and engineering, various real life problems are created as mathematical models before they are solved. These models often lead to special class of ordinary differential equations known as stiff ODEs. A system is regarded as ‘stiff’; if the existing explicit numerical methods fail to efficiently integrate it, or when the step size is determined by the requirements of its stability, rather than the accurateness. The solution of stiff ODEs contains a component with both slowly and rapidly decaying rates due to a large difference in the time scale exhibited by the system. The stiffness property prevents the conventional explicit method from handling the problem efficiently. This nature of stiff ODEs has led to considerable research efforts in developing many implicit mathematical methods. This paper discussed the convergence and order of the current variable step size super-class of block backward differentiation formula (BBDF) for solving stiff initial value problems. The necessary conditions for the convergence of the fourth order variable step size super class of BBDF for solving stiff initial value problems, has been established in this work. It has been shown that the new method is both zero-stable and consistent, which are the requirements for the convergence of any numerical method. The order of the method is also derived to be four. It is therefore concluded that the method is convergent and has significance in solving more complex stiff initial value problems, and could be robustly applied in many fields of study.

Construction of stochastic differential equations of motion in canonical variables

Galiullin proposed a classification of inverse problems of dynamics for the class of ordinary differential equations (ODE). Considered problem belongs to the first type of inverse problems of dynamics (of the three main types of inverse problems of dynamics): the main inverse problem under the additional assumption of the presence of random perturbations. In this paper Hamilton and Birkhoff equations are constructed according to the given properties of motion in the presence of random perturbations from the class of processes with independent increments. The obtained necessary and sufficient conditions for the solvability of the problem of constructing stochastic differential equations of both Hamiltonian and Birkhoffian structure by the given properties of motion are illustrated by the example of the motion of an artificial Earth satellite under the action of gravitational and aerodynamic forces.

Adaptive reachability algorithms for nonlinear systems using abstraction error analysis

In many reachability algorithms for nonlinear ordinary differential equations (ODEs), the tightness of the computed reachable sets mainly depends on abstraction errors and the choice of the set representation. One has to mitigate the resulting wrapping effects by suitable tuning of internally-used algorithm parameters since there exists no wrapping-free algorithm to date. In this work, we investigate the fundamentals governing the abstraction error in reachability algorithms – which we also refer to as set-based solvers – and its dependence on the time step size, leading to the introduction of a gain order. This order is related to measures for local and global abstraction errors and thus relates the well-known concept of convergence order from classical ODE solvers to set-based solvers. Furthermore, the simplification of the set representation is tackled by limiting the Hausdorff distance between the original and reduced sets; we demonstrate this for zonotopes. Both these theoretical advancements are incorporated in a modular adaptive parameter tuning algorithm suited for multiple classes of nonlinear ODEs whose efficiency is demonstrated on a wide range of benchmarks.

Using Dtm To Solve A Special Class Of Ordinary Differential Equations

Using Dtm To Solve A Special Class Of Ordinary Differential Equations