Modelling and simulation of PDE problems in Modelica

Modelica is a new object-oriented language for modeling of large, complex, and heterogeneous physical systems. It is designed to allow for convenient and efficient modeling and simulation of the differential-algebraic equations (DAEs) systems. However, in many engineering fields, partial differential equations (PDEs) are applied. So it is necessary to be able to handle PDE models in Modelica. In this paper, we present how the Modelica language can be extended to support object-oriented modeling with partial differential equations. The description of PDE models in Modelica is introduced, which uses a natural way to describe the PDE models so that the extending language is easier to be understood and more convenient to be used. The prototype implementation included external translator and processor is described. An example modeled with PDE in Modelica and solved using the prototype is presented.

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

On the variational iteration method and other iterative techniques for nonlinear differential equations

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.

In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions.

We propose a new method in which a generative network (GN) set within a reduced-order model (ROM) framework is used to solve inverse problems for partial differential equations (PDE). The aim is to match available measurements and estimate the corresponding uncertainties associated with the states and parameters of a numerical physical simulation. We train the GN using only unconditional simulations of the discretized PDE model. A GN is used here to exploit the ability of these methods to generate realistic outputs from complex probability distributions, such as the ones that represent the possible states and parameters of a physical problem. We compare the proposed method with the gold standard Markov chain Monte Carlo (MCMC). Additionally, we suggest the use of a novel type of time-stepping regularization to improve the representativeness of the physical solution, and we present a new way of evaluating the GN training, taking advantage of the real/generated sample structure. We apply the proposed approaches to a spatio-temporal compartmental model in epidemiology. The results show that the proposed GN-based ROM can efficiently quantify uncertainty and accurately match the measurements and the gold standard. This is achieved using only a limited number of unconditional simulations from the full-order numerical PDE model (40 simulations). The GN-based ROM operates 60 times faster than the gold standard (MCMC), while producing uncertainties that closely match those generated by the MCMC approach. The proposed method is a general framework for quantifying uncertainties in numerical physical simulations and is not restricted to the specific physics of this application.

Comparison of methods for solution of the heat conduction equation with a radiation boundary condition

Integral Equations for Nonlinear Problems in Partial Differential Equations

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Experimental results have shown that anti-cancer therapies, such as radiotherapy and chemotherapy, can modulate the cell cycle and generate cell cycle phase-dependent responses. As a result, obtaining a detailed understanding of the cell cycle is one possible path towards improving the efficacy of many of these therapies. Here, we consider a basic structured partial differential equation (PDE) model for cell progression through the cell cycle, and derive expressions for key quantities, such as the population growth rate and cell phase proportions. These quantities are shown to be periodic and, as such, we compare the PDE model to a corresponding ordinary differential equation (ODE) model in which the parameters are linked by ensuring that the long-term ODE behaviour agrees with the average PDE behaviour. By design, we find that the ODE model does an excellent job of representing the mean dynamics of the PDE model within just a few cell cycles. However, by probing the parameter space we find cases in which this mean behaviour is not a good measure of the PDE population growth. Our analytical comparison of two caricature models (one PDE and one ODE system) provides insight into cases in which the simple ODE model is an appropriate approximation to the PDE model.

An analysis is given of effective strategies for the so-called method of lines solution of the initial boundary-value problem for partial differential equations of the form u/sub t/ = g(x,t,u,u/sub x/,u/sub xx/,f/sub x/), where f = f*x,t,u,u/sub x/,u/sub xx/). Also described is a user-oriented FORTRAN subroutine package called MOL1D for the numerical solution of equations of this type using the method of lines. 5 figures, 10 tables.

Index-analysis for a method of lines discretising multirate partial differential algebraic equations

A further index concept for linear PDAEs of hyperbolic type

The usual solution methods of discretised partial differential equations are based exclusively on matrix-vector multiplications as basis operation. On the one hand, this is supported by the use of sparse matrices (cf. §1.3.2.5); on the other hand, one tries to apply fast iterative methods (e.g., the multigrid method [124], [119, §12]) whose basic steps are matrix-vector multiplications. Krylov methods are based on the same concept. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation1 and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. The $$\mathcal{H}$$ -matrix arithmetic allows the solution of these matrix equations efficiently. Here, the use of hierarchical matrix operations and matrix-valued functions is only one part of the solution method. Another essential fact is that the solution $$ X \in \mathbb{R}^{I \times I}(n==\!\!\!\!\!/ \!\!\!\!/I)$$ can be replaced by an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ . If one considers the equation $$f(X)\;=\;0$$ as a system of n 2 equation for the n 2 components of X, even an optimal solution method has complexity $$\mathcal{O}(n^2)$$ , since this is linear complexity in a number of unknowns (cf. Remark 1.1). Using traditional techniques, the solution of large-scale matrix equations is not feasible. Only an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ with $$\mathcal{O}(n\; \mathrm{log}^*n)$$ data instead of n 2 admits a solution with a cost almost linear with respect to n. Section 15.1 introduces Lyapunov and Sylvester equations and discusses their solution. In Section 15.2 we consider quadratic Riccati equation. An interesting approach uses the matrix version of the sign function from §14.1.1. General nonlinear matrix equations may be solved iteratively by Newton’s method or related methods (cf. Section 15.3). As an example, computing the square root of a positive definite matrix is described in §15.3.1. The influence of the truncation error introduced by H-matrix arithmetic is investigated in Section 15.3.