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The system of partial differential equations which forms the basic semiconductor equations together with appropriate boundary conditions has been investigated and characterized analytically in the previous chapter. This system cannot be solved explicitly in general. Therefore, the solution must be calculated by means of numerical approaches. We shall consider in this chapter such Solution procedures for the sealed equations which read: $$\lambda ^2 \cdot \text{div}\,\text{grad}\,\psi - (n - p - C) = 0$$ (6-1) $$\text{div}(D_n \cdot \text{grad}\,n - \mu _\text{n} \cdot n \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial n}} {{\partial t}}$$ (6-2) $$\text{div}(D_p \cdot \text{grad}\,p - \mu _\text{p} \cdot p \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial p}} {{\partial t}}$$ (6-3) Any numerical approach for the Solution of such a system consists essentially of three tasks. First, the domain, i.e. the Simulation geometry of the device, has to be partitioned into a finite number of subdomains, in which the Solution can be approximated easily with a desired accuracy. Secondly, the differential equations have to be approximated in each of the subdomains by algebraic equations which involve only values of the continuous dependent variables at discrete points in the domain and knowledge of the structure of the chosen funetions which approximate the dependent variables within each of the subdomains. In that way one obtains a fairly large system of, in general nonlinear, algebraic equations with unknowns comprised of approximations of the continuous dependent variables at discrete points. The Solution of this system is the final third task to be carried out. As this problem can be viewed rather independently it will be treated separately in Chapter 7.
10Rational general solutions of systems of first-order algebraic partial differential equations
Rational general solutions of systems of first-order algebraic partial differential equations
12Singular perturbation problems for systems of partial differential equations of elliptic type
Singular perturbation problems for systems of partial differential equations of elliptic type
2On Stability of a Class of Linear Systems with Distributed and Lumped Parameters
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.
Read more12Analysis of dynamic deformation processes with adaptive KALMAN-filtering
The analysis of dynamic deformation processes is an important task in civil and in mechanical engineering. One main emphasis is set on the derivation of the objects transmission function, represented by systems of ordinary or partial differential equations (“white box”-models). The parametric identification of the initial and boundary conditions and relevant material parameters is one of the new challenges in the application of least-square algorithms. Embedding the investigation of the deformation process in a combination of a theoretical and an experimental system analysis suitable methods are provided with adaptive KALMAN-filtering as central identification tool. In this paper the approach of a full system analysis is shown quantifying a dynamic “white-box”-model for the calculation of thermal deformations of bar-shaped machine elements. The task was motivied from mechanical engineering searching new methods for the precise prediction and computational compensation of thermal influences in the heating and cooling phases of machine tools (i.e. robot arms, etc.). The quantification of thermal deformations under variable dynamic loads requires the modelling of the non-stationary spatial temperature distribution inside the object. Based upon FOURIERS law of heat flow the high-grade non-linear temperature gradient is represented by a system of partial differential equations within the framework of a dynamic Finite Element topology. It is shown that adaptive KALMAN-filtering is suitable to quantify relevant disturbance influences and to identify thermal parameters (i.e. thermal diffusivity) with a deviation of only 0,2%. As result an identified (and verified) parametric model for the realistic prediction respectively simu-lation of dynamic temperature processes is presented. Classifying the thermal bend as the main deformation quantity of bar-shaped machine tools, the temperature model is extended to a temperature deformation model. In lab tests with an aluminium column control measurements show that the identified model can be used to predict the columns bend with a mean deviation smaller than 10 mgon. Consequently the deformation model is a precise predictor and suitable for realistic simulations of thermal deformations under variable dynamic loads. In future our activities will be primarily focussed on applications in industrial manufacturing. 1. Parametric identification of dynamic deformation processes The creation of a realistic parametric dynamic deformation model is directly associated with the task to combine systems of differential equations (theoretical physical model) with empirical measurements. Consequently basic principles of system theory can be adapted to the geodetic deformation analysis. The parametric identification of a dynamic “white box”-model 3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006
Read more4A new technique to solve models for the reaction of solid particles and for parallel catalyst deactivation
A new technique to solve models for the reaction of solid particles and for parallel catalyst deactivation
On the solvability of the boundary value problem for one class of nonlinear systems of high-order partial differential equations
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.
Read moreOn a Coupled System of Partial Differential Equations
Boundary value problems are considered for a system of stationary partial differential equations that stems from the models describing the implantation of impurities into semiconductor devices and is similar to the equations of carrier transport. The existence of solutions by functional analytic means (energetic method) is proved. Besides, an upper estimate for the set of solutions is obtained.
Read more1On the theory and practice of thin-walled structures
The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.
Read more14Local Transformations and Conservation Laws
A continuous symmetry of a system of partial differential equations (PDEs) is a transformation that leaves invariant the solution manifold of the system, i.e, it maps (deforms) any solution of the system into a solution of the same system. This definition is topological in nature. However, in practice, the direct calculation of the continuous symmetries of a given system of PDEs restricts one to consider symmetries that are local transformations acting on its space of independent variables, dependent variables and their derivatives. Lie’s algorithm to determine Lie groups of point transformations (point symmetries) of differential equations was presented in Bluman & Anco (2002) [see also Ovsiannikov [(1962), (1982)]; Bluman & Cole (1974); Olver (1986); Bluman & Kumei (1989); Stephani (1989); Hydon (2000); Cantwell (2002)]. Point symmetries arise from solutions of linear systems of determining equations for components of infinitesimal generators for the independent and dependent variables of a given PDE system, where these components themselves depend only on the given PDE system’s independent and dependent variables. Point transformations acting on the space of the given independent and dependent variables of a given PDE system can be extended (prolonged) to point transformations acting on the space of the given independent variables, dependent variables, and their derivatives to any finite order.
Read moreContact geometry of second order I
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.
Read more140Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations
Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations
2One method for solving systems of nonlinear partial differential equations
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.).
Read more2Reliability-Oriented Modeling of Bellows Compensators: A Comparative PDE-Based Study Using Finite Difference and Finite Element Methods
Bellows compensators are critical components in pipeline systems, designed to absorb thermal expansions, vibrations, and pressure reflections. Ensuring their operational reliability requires accurate prediction of the stress–strain state (SSS) and stability under internal pressure. This study presents a comprehensive mathematical model for analyzing corrugated bellows compensators, formulated as a boundary value problem for a system of partial differential equations (PDEs) within the Kirchhoff–Love shell theory framework. Two numerical approaches are developed and compared: a finite difference method (FDM) applied to a reduced axisymmetric formulation to ordinary differential equations (ODEs) and a finite element method (FEM) for the full variational formulation. The FDM scheme utilizes a second-order implicit symmetric approximation, ensuring stability and efficiency for axisymmetric geometries. The FEM model, implemented in Ansys 2020 R2, provides high fidelity for complex geometries and boundary conditions. Convergence analysis confirms second-order spatial accuracy for both methods. Numerical experiments determine critical pressures based on the von Mises yield criterion and linearized buckling analysis, revealing the influence of geometric parameters (wall thickness, number of convolutions) on failure mechanisms. The results demonstrate that local buckling can occur at lower pressures than that of global buckling for thin-walled bellows with multiple convolutions, which is critical for structural reliability assessment. The proposed combined approach (FDM for rapid preliminary design and FEM for final verification) offers a robust and efficient methodology for bellows design, enhancing reliability and reducing development time. The work highlights the importance of integrating rigorous PDE-based modeling with modern numerical techniques for solving complex engineering problems with a focus on structural integrity and long-term performance.
Read moreError analysis of tailor-made time integration schemes for certain classes of wave-type equations
This thesis is concerned with the time integration of three classes of wave-type partial differential equations. Each of these equations has numerical challenges, including nonlocal-in-time material laws, nontrivial boundary conditions, and an unbounded spatial domain. We construct tailor-made schemes and provide a rigorous numerical analysis. We consider the semiclassical magnetic Schrödinger equation on the full space $\mathbb{R}^d$ with possibly time-dependent magnetic and electric potentials. For the approximation we use an appropriate Gaussian ansatz function, which leads to a set of ordinary differential equations. We show that in a special case, this ansatz function is the exact solution to the magnetic Schrödinger equation. Furthermore, we provide error bounds with respect to the semiclassical parameter in $L^2$-norm and we improve the error bound for relevant physical quantities of interest. The second setting is concerned with a scattering problem for \MaxEq~on an unbounded domain. The scatterer consists of a nonlocal-in-time material, which is modeled as a convolution in \MaxEq. In this situation, we use a coupling on the scatterer's bounded interface and derive a boundary integral equation. For the discretization we employ a convolution quadrature combined with a boundary element method. Finally, we show that this numerical approach is stable and we prove convergence rates. For a semilinear viscoacoustic wave equation with a retarded material law and kinetic boundary conditions, we construct and analyze an implicit-explicit (IMEX) scheme. The IMEX scheme is computationally efficient, since it avoids the solution of nonlinear systems. The material law is described by a convolution term with exponential kernels. After applying an appropriate shift, we couple the convolution term as an auxiliary variable to the first-order system of partial differential equations. For the kinetic boundary conditions we make use of suitable bulk-surface Sobolev spaces to show wellposedness.
Read more2A Galerkin procedure for systems of differential equations
A continuous time and an extrapolated coefficient Crank-Nicolson-Galerkin method are considered for approximating solutions of boundary and initial value problems for a quasi-linear parabolic system of partial differential equations which is coupled to a non-linear system of ordinary differential equations. A priori bounds are derived that reduce the estimation of error to problems in approximation theory. Then approximation theory results yield optimal order rates of convergence for theH 1 (Ω) norm. The extrapolated coefficient method yields linear algebraic equations for strongly non-linear problems.
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