Reduction of Linear Systems of ODEs with Optimal Replacement Variables

The purpose of this work is to propose and demonstrate a way to explicitly transform polynomial ODE systems to linear ODE systems. With the help of an additional first integral, the one-dimensional Riccati equation is transformed to a linear system of three ODEs with variable coefficients. Solving the system, we can find a solution to the original Riccati equation in the general form or only to the Cauchy problem. The Riccati equation is one of the most interesting nonlinear first order differential equations. It is proved that there is no general solution of the Riccati equation in the form of quadratures; however, if at least one particular solution is known, then its general solution is also found. Thus, it is enough only to find a particular solution of the linear system of ODEs. The applied transformation method is a special case of the method described in our work [Zaytsev M. L., Akkerman V. B. (2020) On the identification of solutions to Riccati equation and the other polynomial systems of ODEs // preprint, Research Gate. DOI: 10.13140 / RG.2.2.26980.60807]. This method uses algebraic transformations and transition to new unknowns consisting of products of the original unknowns. The number of new unknowns becomes less than the number of equations. For the multidimensional Riccati equations, we do not present the corresponding linear system of ODEs because of the large number of linear equations obtained (more than 100). However, we present the first integral with which this can be done. In this paper, we also propose a method for finding the first integral, which can be used to reduce a search for the solution of any polynomial systems of ODEs to a search of solutions to linear systems of ODEs. In particular, if the coefficients in these equations are constant, then the solution is found explicitly.

Known mathematical packages, MATLAB, Maple, Mathematica, MathCAD and others may get wrong, often plausible, the result of numerical solution of ODE systems with low, given by default, the requirements for mathematical accuracy of the results of the numerical solution of ODE systems. Since ODE system parameters obtained usually experimental, with a low mathematical precision, so the requirements for the precision of the results of mathematical solutions of ODE systems is low (for example, in MATLAB package required precision is 0.001). The article offers a basic set of tests to assess the range of applicability of the relevant solvers. The basic set of test problems for ODE solvers systems include linear ODE systems with the known analytic solution and nonlinear systems with known graphics solution. Presented comparative results of solutions for proposed problems using MATLAB solvers and manzhuk program from a library of standard mathematical programs SADEL (Sets of Algebraic and Differential Equations solvers Library), which has been designed for reliable and accurate solving of systems of linear algebraic equations (LAE) and ODE systems. The results can be used in mathematical modeling of dynamic systems described by ODE systems.

Adaptation of homotopy-perturbation method for numeric–analytic solution of system of ODEs

This paper is about the finite-time steady-state to steady-state transfer of systems governed by boundary controlled 1D heat and beam equations. We address this problem using semi-discretization and flatness. We derive an n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> -order semi-discretized system of linear ODEs in time for the PDE system (which is either a heat equation or a beam equation) by approximating the spatial derivatives in the PDE system using Taylor's theorem. We show, under some regularity assumptions on the boundary input f and the initial state, that the state trajectory of the ODE systems converge to the state trajectory of the PDE system that they approximate as the number of discretization steps n goes to infinity. Given a T > 0 and an initial steady-state u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> at time t = 0 and a final steady-state u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> at time t = T for the PDE system, we construct an input f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> for the n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> -order semi-discrete ODE system to transfer its state from discretized u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> to discretized u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> in time T. This f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is constructed using the flatness approach. We show that the sequence (f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">=</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> converges to a limiting function f as n → ∞ and f is the input that transfers the PDE system from u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> to u <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> . We illustrate our results using numerical simulations.

A combined semi-discretized spectral matrix collocation algorithm based on (new) family of Krawtchouk polynomials is proposed to solve the time-dependent nonlinear auto-catalytic glycolysis reaction-diffusion system arising in mathematical chemistry. The first stage of the numerical algorithm is devoted to the Taylor series time advancement procedure yielding to a (linear and steady) system of ODEs. In the second stage and in each time frame, a matrix collocation technique based on the Krawtchouk polynomials is utilized to the resulting system of ODEs in an iterative manner. The results of the performed numerical experiments with Neumann boundary conditions are given to show the utility and applicability of the combined Taylor-Krawtchouk spectral collocation algorithm. The positive property of the glycolysis chemical model is sustained by the proposed algorithm and verified through comparisons with existing numerical methods in the literature. The combined technique is simple and flexible enough to easily produce the approximate solutions of diverse physical and applied models in engineering and science.

Based on the multilevel interpolation theory, we constructed a meshless adaptive multiscale interpolation operator (MAMIO) with the radial basis function. Using this operator, any nonlinear partial differential equations such as Burgers equation can be discretized adaptively in physical spaces as a nonlinear matrix ordinary differential equation. In order to obtain the analytical solution of the system of ODEs, the homotopy analysis method (HAM) proposed by Shijun Liao was developed to solve the system of ODEs by combining the precise integration method (PIM) which can be employed to get the analytical solution of linear system of ODEs. The numerical experiences show that HAM is not sensitive to the time step, and so the arithmetic error is mainly derived from the discrete in physical space.

The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation–oscillations

This paper proposes a methodology for output feedback control of parabolic PDE systems with input constraints. Initially, Galerkin's method is used for the derivation of a finite-dimensional ODE system that captures the dominant dynamics of the PDE system. This ODE system is then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded output feedback control laws that use only measurements of the outputs and provide, at the same time, an explicit characterization of the set of admissible control actuator locations that can be used to guarantee closed-loop stability for a given initial condition. Precise conditions that guarantee stability of the constrained closed-loop parabolic PDE system are provided. The proposed output feedback design is shown to recover, asymptotically, the set of stabilizing actuator locations obtained under state feedback, as the separation between the fast and slow eigenvalues of the spatial differential operator increases.

The current investigations are based on a numerical analysis of the role of Casson fluid with Hall forces between two parallel disks. Thermal performance and cooling process can be characterized by utilizing a ternary nanofluid model in two disks. The concentration of nanofluid and nanoparticles is measured using a variation of heat sink and thermal radiation (quadratic). Thermal energy occurs considering nonlinear-thermal radiation. The current model boosts energy conversion rates and overall performance in concentrated solar power plants by improving heat transfer. Additionally, these nanofluids are essential to photovoltaic panel cooling systems, helping to reduce temperature-related efficiency losses and increase panel lifetime. The model is based on a system of ODEs and is obtained via transformations through a system of PDEs. Based on the literature review, it was estimated that the current model is not yet solved by utilizing the finite element approach. Computational examples include hybrid nano-martial ( Cu - Ag /Blood) and ternary hybrid nanomaterial ( Cu - Ag - Fe 3 O 4 /Blood) underlying, velocities in both directions, heat energy, and concentration profiles. It was observed that the hybrid nanomaterial significantly achieves less motion compared to the ternary hybrid nano-material, whilst considerable thickness and width are achieved for the ternary hybrid nano-material compared to the hybrid nanomaterial. Heat energy production is improved with the radiation, magnetic, and heat source numbers. For temperature distribution, the performance of hybrid nano-martial is lower than that of ternary hybrid nano-martial. Also, it is observed that thermal transportation production is intensified for ternary hybrid nano-martial compared to hybrid nano-martial.

Time-delays are frequently a source of instability, but for some systems, the presence of delay may have a stabilizing effect. A time-delay approach to sampled-data control, where the system is modeled as a continuous-time system with the delayed control input became popular in networked control systems, where the plant and the controller exchange data via communication network. In the present talk delay effects on stability, positivity and control will be discussed. A time-delay approach to sampled-data and network-based control of ODE and PDE systems will be presented, where variable sampling intervals, communication delays and scheduling protocols are taken into account. Differently from other approaches, this approach allows communication delays larger than the sampling intervals in the presence of scheduling protocols. As an application of PDE results, a network-based deployment of multi-agent systems via PDEs will be considered. Finally a very recent time-delay approach, this time to averaging, will be presented. Here the time-delay approach provides constructive upper bounds on the small parameter that preserve the stability.

For a linear system of singular ordinary differential equations, necessary and suffcient conditions are established for the existence of a unique C^m -smooth solution. A reduction to cordial Volterra integral equations is used.

Explicit Wei–Norman formulae for matrix Lie groups via Putzer's method

This research article explains the Joule heating and inclined magnetic field effects on Ti6Al4V-MWCNT/[Formula: see text] hybrid nanofluids toward an exponentially shrinking surface. The system of flow governing equations, which is a coupled system of non-linear PDE’s, are transformed into a system of non-dimensional ODE’s by using suitable similarity transformations. The shooting method along with a fourth order Runge-Kutta approach is used to solve the system of resultant ODE’s. As a result, the dual solutions are identified for this problem. By utilizing the least eigenvalue approach, the first solution is found to be realistic and stable whereas the second solution is not stable. In order to verify the accuracy of outcomes, it is compared with the previously published results, and it is found that the present analysis is accurate upto four decimal places. Further, the streamline profiles are also provided in addition to the least eigenvalue profiles to understand the flow patterns. Finally, the flow separation point is identified. The velocity profiles improves in the case of first solution but it declines in the case of second solution with a rise in the shrinking strength parameter, however the temperature profiles behave in opposite manner. The skin friction coefficient and heat transfer rate of Ti6Al4V-MWCNT/[Formula: see text] based hybrid nanofluids are found to be better with higher values of inclined MHD [Formula: see text] and incline angle [Formula: see text].

Homoclinic Bifurcation in an SIQR Model for Childhood Diseases

Weak well-posedness for a modified two-component Camassa–Holm system