Deep-MacroFin: Informed Equilibrium Neural Network for Continuous-Time Economic Models

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

Many problems of science and technique lead to partial differential equations. The mathematical theories of such equations, especially of the nonlinear ones, are very intricate, such that their numerical study becomes inevitable. To classify the partial differential equations, this chapter uses various criteria, that is, considering the order of the derivatives (first order, second order, or nth order), the linearity character (linear, quasilinear, or nonlinear equations), the influence of the integration domain at a point (equations of elliptic, parabolic, or hyperbolic type), and the types of limit conditions (Dirichlet, Neumann, or mixed problems). It presents the numerical methods for the integration of partial differential equations and of systems of partial differential equations. The numerical methods are point matching method, Ritz's method, Galerkin's method, and method of the least squares. Finally, the chapter gives numerical examples and some applications.

A design method of finite dimensional nonlinear disturbance observer based control (NLDOBC) is proposed for a class of nonlinear dissipative partial differential equation (PDE) systems, where the disturbance is modeled by an exosystem of nonlinear ordinary differential equations (ODEs). Motivated by the fact that the dominant structure of the dissipative PDE is usually characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to give a slow subsystem of low dimensional nonlinear ODEs. By using the nonlinear slow subsystem and exosystem, a nonlinear disturbance observer (NLDO) is constructed to estimate the disturbance. Then, on the basis of NLDO, an NLDOBC method is proposed in terms of linear matrix inequality (LMI) to ensure the exponential stability of the closed-loop PDE system in the presence of the modeled disturbance. Finally, the simulation results on control of one dimensional Fisher diffusion-reaction system show the effectiveness of the proposed method.

In this article, we develop an invariant subspace method for a system of time-fractional nonlinear partial differential equations in $$(1+2)$$ dimensions. Efficacy of the method is demonstrated by solving coupled system of nonlinear time-fractional diffusion equations and coupled system of time-fractional Burger’s equations in higher dimensions. Furthermore, the algorithmic approach to find more than one invariant subspace is proposed and corresponding exact solutions are constructed.

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.

A new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients

A new technique to solve models for the reaction of solid particles and for parallel catalyst deactivation

This article studies the problem of prescribed‐time constrained tracking control for a class of hyperbolic partial differential equation (PDE) systems with actuator dynamics, which are described by a set of nonlinear ordinary differential equations (ODEs). Since the control input only appears in the ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite challenging. The most important is that for the control of such ODE controlled PDE systems we mainly make the following two contributions: (1) the controlled output of the PDE system tracks the reference signal within the prescribed time; (2) the controlled output and all the actuator states are constrained. It is the first time that such a prescribed‐time constrained tracking control problem is addressed for the PDE‐ODE coupled system considered in this article. Through rigorous theoretical proof, it is demonstrated that all the system states and control signals are bounded and sufficiently continuous by configuring appropriate design parameters. Finally, the performance is investigated via numerical simulation.

Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations