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Combined methods for solving time-varying semilinear differential-algebraic equations with the use of spectral projectors and applications

Maria Filipkovska

Abstract

Two combined methods for computing solutions of time-varying semilinear differential-algebraic equations (descriptor systems) are obtained. When constructing the methods, time-varying spectral projectors which can be found numerically are used. This enables one to numerically solve the differential-algebraic equation (DAE) in the original form without additional analytical transformations. The convergence and correctness of the developed methods are proved. The methods are applicable to the semilinear DAEs with the continuous nonlinear part which may not be differentiable in time. The global Lipschitz condition and other conditions of this kind are not used in the presented theorems on the global solvability of DAEs and on the convergence of the methods. This extends the scope of the methods. The obtained theorems ensure both the existence of a unique global exact solution and the convergence of the methods, which enables one to compute an approximate solution on any given time interval. Numerical examples illustrating the capabilities of the methods and their effectiveness in various situations are provided. To demonstrate the practical application of the obtained methods and theorems, the numerical and theoretical analyses of mathematical models of the dynamics of electric circuits are carried out. It is shown that their results are consistent.

Combined methods for solving time-varying semilinear differential-algebraic equations with the use of spectral projectors and applications

Abstract

Two combined methods for computing solutions of time-varying semilinear differential-algebraic equations (descriptor systems) are obtained. When constructing the methods, time-varying spectral projectors which can be found numerically are used. This enables one to numerically solve the differential-algebraic equation (DAE) in the original form without additional analytical transformations. The convergence and correctness of the developed methods are proved. The methods are applicable to the semilinear DAEs with the continuous nonlinear part which may not be differentiable in time. The global Lipschitz condition and other conditions of this kind are not used in the presented theorems on the global solvability of DAEs and on the convergence of the methods. This extends the scope of the methods. The obtained theorems ensure both the existence of a unique global exact solution and the convergence of the methods, which enables one to compute an approximate solution on any given time interval. Numerical examples illustrating the capabilities of the methods and their effectiveness in various situations are provided. To demonstrate the practical application of the obtained methods and theorems, the numerical and theoretical analyses of mathematical models of the dynamics of electric circuits are carried out. It is shown that their results are consistent.
Paper Structure (24 sections, 11 theorems, 147 equations, 14 figures, 2 tables)

This paper contains 24 sections, 11 theorems, 147 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Index of a regular pencil of operators, spectral projectors and their application
  4. Index of a regular pencil $\lambda A(t)+B(t)$
  5. Spectral projectors of the Riesz type and their application
  6. Method of the spectral projectors for the reduction of a time-varying semilinear DAE to the system of an explicit ODE and an algebraic equation.
  7. Existence, uniqueness and boundedness of global solutions of time-varying semilinear DAEs
  8. The existence, uniqueness and boundedness of global solutions of DAEs \ref{['DAE']} and \ref{['DAE1']}.
  9. Combined numerical methods: the construction, convergence and orders
  10. Calculation of the spectral projectors and preliminaries
  11. Method 1 (the simple combined method)
  12. Method 2 (the combined method with recalculation)
  13. Numerical experiments
  14. Example 1: Analysis of a mathematical model of the electrical circuit dynamics
  15. Theoretical analysis of the mathematical model of the electrical circuit dynamics
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $f\in C([t_+,\infty)\times {\mathbb R}^n, {\mathbb R}^n)$, $\dfrac{\partial f}{\partial x} \in C([t_+,\infty)\times {\mathbb R}^n, \mathrm{L}({\mathbb R}^n))$, $A, B\in C^1([t_+,\infty),\mathrm{L}({\mathbb R}^n))$, the pencil $\lambda A(t)+B(t)$ satisfy index1, where $C_2\in C^1([t_+,\infty),(0 Then for each initial point $(t_0,x_0)\in L_{t_+}$ there exists a unique global solution of the IVP

Figures (14)

  • Figure 1: The example of a Lagrange-stable solution: The plots of the components $x_1(t)$, $x_2(t)$, $x_3(t)$ of the approximating solution $x(t)=(x_1(t),x_2(t),x_3(t))^{\mathop{\mathrm{T}}}$ computed for the DAE \ref{['DAE']}, \ref{['NestCoefDAE']} with the functions \ref{['Ex2_param']}, \ref{['StepenFunc2']}, where $a=b=1$, $k=m=2$, and the initial values $t_0=0$, $x_0=(0,0,0)^{\mathop{\mathrm{T}}}$. The analysis of the presented graphs shows that the solution exists on the given interval and its norm does not increase with increasing time. When the interval is increased by a factor of 10 and more, the qualitative picture of the behavior of the numerical solution does not change (therefore, the corresponding graphs were not presented here). Thus, the results of the numerical experiment are consistent with the conclusion about the Lagrange stability of the DAE, which was obtained using the corresponding theorem.
  • Figure 2: The example of a global solution: The plots of the components $x_1(t)$, $x_2(t)$ and $x_3(t)$ of the approximating solution $x(t)=(x_1(t),x_2(t),x_3(t))^{\mathop{\mathrm{T}}}$ computed (by method 1) for the DAE \ref{['DAE']}, \ref{['NestCoefDAE']} with the functions \ref{['Ex1_1-3_param']} and \ref{['StepenFunc2']}, where $a=b=1$, $k=m=2$, and the initial values $t_0=0$ and $x_0=(0,37,3)^{\mathop{\mathrm{T}}}$. In this case, the exact solution is global, but it can be unbounded on $[t_0,\infty)$. This is because the conditions for the global solvability of the DAE \ref{['DAE']}, \ref{['NestCoefDAE']}, specified in Section \ref{['TheorAnalPIMM']}, hold, but the additional conditions for the Lagrange stability are not fulfilled. The presented graphs demonstrate the same behavior pattern of the solution. When the interval is increased by a factor of 10, the qualitative picture of the behavior of the solution does not change.
  • Figure 3: The plot of $U(t)$
  • Figure 4: The example of a solution for the case when the function $U(t)$ is continuous, but not differentiable: The plots of the components $x_1(t)$, $x_2(t)$, $x_3(t)$ of the approximate solution $x(t)=(x_1(t),x_2(t),x_3(t))^{\mathop{\mathrm{T}}}$ computed for the DAE \ref{['DAE']}, \ref{['NestCoefDAE']} with the functions \ref{['Upila-func']}, \ref{['ExPila_param']} and \ref{['StepenFunc2']}, where $a=3$, $b=4$, $k=2$, $m=2$, and the initial values $t_0=0$, $x_0=(0,0,0)^{\mathop{\mathrm{T}}}$. The theoretical analysis shows that the exact solution is Lagrange stable and the presented plots demonstrate the same behavior pattern of the approximate solution.
  • Figure 5: The electric circuit diagram
  • ...and 9 more figures

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 2.1: Global solvability of the DAE \ref{['DAE']} Fil.DE-1
  • Proposition 2.1
  • proof
  • Theorem 2.2: Global solvability of the DAE \ref{['DAE']}, Fil.DE-1
  • Remark 2.1
  • Proposition 2.2: Fil.DE-1
  • Theorem 2.3: Lagrange stability of the DAE \ref{['DAE']} Fil.DE-1
  • Corollary 2.1
  • Remark 2.2
  • ...and 11 more