Дисертації: "Singularly Perturbed Differential Equation"

1

Mbroh, Nana Adjoah. "On the method of lines for singularly perturbed partial differential equations." University of the Western Cape, 2017. http://hdl.handle.net/11394/5679.

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Magister Scientiae - MSc
Many chemical and physical problems are mathematically described by partial differential equations (PDEs). These PDEs are often highly nonlinear and therefore have no closed form solutions. Thus, it is necessary to recourse to numerical approaches to determine suitable approximations to the solution of such equations. For solutions possessing sharp spatial transitions (such as boundary or interior layers), standard numerical methods have shown limitations as they fail to capture large gradients. The method of lines (MOL) is one of the numerical methods used to solve PDEs. It proceeds by the discretization of all but one dimension leading to systems of ordinary di erential equations. In the case of time-dependent PDEs, the MOL consists of discretizing the spatial derivatives only leaving the time variable continuous. The process results in a system to which a numerical method for initial value problems can be applied. In this project we consider various types of singularly perturbed time-dependent PDEs. For each type, using the MOL, the spatial dimensions will be discretized in many different ways following fitted numerical approaches. Each discretisation will be analysed for stability and convergence. Extensive experiments will be conducted to confirm the analyses.

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2

Song, Xuefeng. "Dynamic modeling issues for power system applications." Texas A&M University, 2003. http://hdl.handle.net/1969.1/1591.

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Power system dynamics are commonly modeled by parameter dependent nonlinear differential-algebraic equations (DAE) x p y x f ) and 0 = p y x g ) . Due to (,, (,, the algebraic constraints, we cannot directly perform integration based on the DAE. Traditionally, we use implicit function theorem to solve for fast variables y to get a reduced model in terms of slow dynamics locally around x or we compute y numerically at each x . However, it is well known that solving nonlinear algebraic equations analytically is quite difficult and numerical solution methods also face many uncertainties since nonlinear algebraic equations may have many solutions, especially around bifurcation points. In this thesis, we apply the singular perturbation method to model power system dynamics in a singularly perturbed ODE (ordinary-differential equation) form, which makes it easier to observe time responses and trace bifurcations without reduction process. The requirements of introducing the fast dynamics are investigated and the complexities in the procedures are explored. Finally, we propose PTE (Perturb and Taylors expansion) technique to carry out our goal to convert a DAE to an explicit state space form of ODE. A simplified unreduced Jacobian matrix is also introduced. A dynamic voltage stability case shows that the proposed method works well without complicating the applications.

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3

Iragi, Bakulikira. "On the numerical integration of singularly perturbed Volterra integro-differential equations." University of the Western Cape, 2017. http://hdl.handle.net/11394/5669.

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Magister Scientiae - MSc
Efficient numerical approaches for parameter dependent problems have been an inter- esting subject to numerical analysts and engineers over the past decades. This is due to the prominent role that these problems play in modeling many real life situations in applied sciences. Often, the choice and the e ciency of the approaches depend on the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These singularly perturbed problems (SPPs) are governed by integro-differential equations in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches zero, the solution undergoes fast transitions across narrow regions of the domain (termed boundary or interior layer) thus affecting the convergence of the standard numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical methods. This work seeks to investigate some "numerical methods that have been used to solve SPVIDEs. It also proposes alternative ones. The various numerical methods are composed of a fitted finite difference scheme used along with suitably chosen interpolating quadrature rules. For each method investigated or designed, we analyse its stability and convergence. Finally, numerical computations are carried out on some test examples to con rm the robustness and competitiveness of the proposed methods.

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4

Davis, Paige N. "Localised structures in some non-standard, singularly perturbed partial differential equations." Thesis, Queensland University of Technology, 2020. https://eprints.qut.edu.au/201835/1/Paige_Davis_Thesis.pdf.

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This thesis addresses the existence and stability of localised solutions in some nonstandard systems of partial differential equations. In particular, it locates the linearised spectrum of a Keller-Segel model for bacterial chemotaxis with logarithmic chemosensitivity, establishes the existence of travelling wave solutions to the Gatenby-Gawlinski model for tumour invasion with the acid-mediation hypothesis using geometric singular perturbation theory, and formulates the Evans function for a trivial defect solution in a general reaction diffusion equation with an added heterogeneous defect. Extending the analysis to these non-standard problems provides a foundation and insight for more general dynamical systems.

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5

Adkins, Jacob. "A Robust Numerical Method for a Singularly Perturbed Nonlinear Initial Value Problem." Kent State University Honors College / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=ksuhonors1513331499579714.

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6

Höhne, Katharina. "Analysis and numerics of the singularly perturbed Oseen equations." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-188322.

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Be it in the weather forecast or while swimming in the Baltic Sea, in almost every aspect of every day life we are confronted with flow phenomena. A common model to describe the motion of viscous incompressible fluids are the Navier-Stokes equations. These equations are not only relevant in the field of physics, but they are also of great interest in a purely mathematical sense. One of the difficulties of the Navier-Stokes equations originates from a non-linear term. In this thesis, we consider the Oseen equations as a linearisation of the Navier-Stokes equations. We restrict ourselves to the two-dimensional case. Our domain will be the unit square. The aim of this thesis is to find a suitable numerical method to overcome known instabilities in discretising these equations. One instability arises due to layers of the analytical solution. Another instability comes from a divergence constraint, where one gets poor numerical accuracy when the irrotational part of the right-hand side of the equations is large. For the first cause, we investigate the layer behaviour of the analytical solution of the corresponding stream function of the problem. Assuming a solution decomposition into a smooth part and layer parts, we create layer-adapted meshes in Chapter 3. Using these meshes, we introduce a numerical method for equations whose solutions are of the assumed structure in Chapter 4. To reduce the instability caused by the divergence constraint, we add a grad-div stabilisation term to the standard Galerkin formulation. We consider Taylor-Hood elements and elements with a discontinous pressure space. We can show that there exists an error bound which is independent of our perturbation parameter and get information about the convergence rate of the method. Numerical experiments in Chapter 5 confirm our theoretical results.

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7

Reibiger, Christian. "Optimal Control Problems with Singularly Perturbed Differential Equations as Side Constraints: Analysis and Numerics." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-162862.

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It is well-known that the solution of a so-called singularly perturbed differential equation exhibits layers. These are small regions in the domain where the solution changes drastically. These layers deteriorate the convergence of standard numerical algorithms, such as the finite element method on a uniform mesh. In the past many approaches were developed to overcome this difficulty. In this context it was very helpful to understand the structure of the solution - especially to know where the layers can occur. Therefore, we have a lot of analysis in the literature concerning the properties of solutions of such problems. Nevertheless, this field is far from being understood conclusively. More recently, there is an increasing interest in the numerics of optimal control problems subject to a singularly perturbed convection-diffusion equation and box constraints for the control. However, it is not much known about the solutions of such optimal control problems. The proposed solution methods are based on the experience one has from scalar singularly perturbed differential equations, but so far, the analysis presented does not use the structure of the solution and in fact, the provided bounds are rather meaningless for solutions which exhibit boundary layers, since these bounds scale like epsilon^(-1.5) as epsilon converges to 0. In this thesis we strive to prove bounds for the solution and its derivatives of the optimal control problem. These bounds show that there is an additional layer that is weaker than the layers one expects knowing the results for scalar differential equation problems, but that weak layer deteriorates the convergence of the proposed methods. In Chapter 1 and 2 we discuss the optimal control problem for the one-dimensional case. We consider the case without control constraints and the case with control constraints separately. For the case without control constraints we develop a method to prove bounds for arbitrary derivatives of the solution, given the data is smooth enough. For the latter case we prove bounds for the derivatives up to the second order. Subsequently, we discuss several discretization methods. In this context we use special Shishkin meshes. These meshes are piecewise equidistant, but have a very fine subdivision in the region of the layers. Additionally, we consider different ways of discretizing the control constraints. The first one enforces the compliance of the constraints everywhere and the other one enforces it only in the mesh nodes. For each proposed algorithm we prove convergence estimates that are independent of the parameter epsilon. Hence, they are meaningful even for small values of epsilon. As a next step we turn to the two-dimensional case. To be able to adapt the proofs of Chapter 2 to this case we require bounds for the solution of the scalar differential equation problem for a right hand side f only in W^(1,infty). Although, a lot of results for this problem can be found in the literature but we can not apply any of them, because they require a smooth right hand side f in C^(2,alpha) for some alpha in (0,1). Therefore, we dedicate Chapter 3 to the analysis of the scalar differential equations problem only using a right hand side f that is not very smooth. In Chapter 4 we strive to prove bounds for the solution of the optimal control problem in the two dimensional case. The analysis for this problem is not complete. Especially, the characteristic layers induce subproblems that are not understood completely. Hence, we can not prove sharp bounds for all terms in the solution decomposition we construct. Nevertheless, we propose a solution method. Numerical results indicate an epsilon-independent convergence for the considered examples - although we are not able to prove this.

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8

Kaiser, Klaus [Verfasser], Sebastian [Akademischer Betreuer] Noelle, Jochen [Akademischer Betreuer] Schütz, and Claus-Dieter [Akademischer Betreuer] Munz. "A high order discretization technique for singularly perturbed differential equations / Klaus Kaiser ; Sebastian Noelle, Jochen Schütz, Claus-Dieter Munz." Aachen : Universitätsbibliothek der RWTH Aachen, 2018. http://d-nb.info/1187251372/34.

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9

Reibiger, Christian [Verfasser], Hans-Görg [Akademischer Betreuer] Roos, and Gert [Akademischer Betreuer] Lube. "Optimal Control Problems with Singularly Perturbed Differential Equations as Side Constraints: Analysis and Numerics / Christian Reibiger. Gutachter: Hans-Görg Roos ; Gert Lube. Betreuer: Hans-Görg Roos." Dresden : Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://d-nb.info/106909658X/34.

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10

Roos, Hans-Görg, and Martin Schopf. "Layer structure and the galerkin finite element method for a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales." Cambridge University Press, 2015. https://tud.qucosa.de/id/qucosa%3A39046.

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We consider a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales. Based on sharp estimates for first order derivatives, Linß [T. Linß, Computing 79 (2007) 23–32.] analyzed the upwind finite-difference method on a Shishkin mesh. We derive such sharp bounds for second order derivatives which show that the coupling generates additional weak layers. Finally, we prove the first robust convergence result for the Galerkin finite element method for this class of problems on modified Shishkin meshes introducing a mesh grading to cope with the weak layers. Numerical experiments support our theory.

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11

Gordon, Brandon W. (Brandon William). "State space modeling of differential-algebraic systems using singularly perturbed sliding manifolds." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/9340.

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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1999.
Includes bibliographical references (p. 126-128).
This thesis introduces a new approach for modeling and control of algebraically constrained dynamic systems. The formulation of dynamic systems in terms of differential equations ·and algebraic constraints provides a systematic framework that is well suited for object oriented modeling of thermo-fluid systems. In this approach, differential equations are used to describe the evolution of subsystem states and algebraic equations are used to define the interconnections between the subsystems (boundary conditions). Algebraic constraints also commonly occur as a result of modeling simplifications such as steady state approximation of fast dynamics and rigid body assumptions that result in kinematic constraints. Important examples of algebraically constrained dynamic systems include multi-body problems, chemical processes, and two phase thermo-fluid systems. Differential-algebraic equation (DAE) systems often referred to as descriptor, implicit, or singular systems present a number of difficult problems in simulation and control. One of the key difficulties is that DAEs are not expressed in an explicit state space form required by many simulation and control design methods. This is particularly true in control of nonlinear DAE systems for which there are few known results. Existing control methods for nonlinear DAEs have so far relied on deriving state space models for limited classes of problems. A new approach for state space modeling of DAEs is developed by formulating an equivalent nonlinear control problem. The zero dynamics of the control system represent the dynamics of the original DAE. This new connection between DAE model representation and nonlinear control is used to obtain state space representations for a general class of differential-algebraic systems. By relating nonlinear control concepts to DAE structural properties a sliding manifold is constructed that asymptotically satisfies the constraint equations. Sliding control techniques are combined with elements of singular perturbation theory to develop an efficient state space model with properties necessary for controller synthesis. This leads to the singularly perturbed sliding manifold (SPSM) approach for state space realization. The new approach is demonstrated by formulating a state space model of vapor compression cycles. This allows verification of the method and provides more insight into the problems associated with modeling differential algebraic systems.
by Brandon W. Gordon.
Ph.D.

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12

Kunert, Gerd. "A note on the energy norm for a singularly perturbed model problem." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100062.

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A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.

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13

Maddah, Sumayya Suzy. "Formal reduction of differential systems : Singularly-perturbed linear differential systems and completely integrable Pfaffian systems with normal crossings." Thesis, Limoges, 2015. http://www.theses.fr/2015LIMO0065/document.

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Dans cette thèse, nous nous sommes intéressés à l'analyse locale de systèmes différentiels linéaires singulièrement perturbés et de systèmes de Pfaff complètement intégrables et multivariés à croisements normaux. De tels systèmes ont une vaste littérature et se retrouvent dans de nombreuses applications. Cependant, leur résolution symbolique est toujours à l'étude. Nos approches reposent sur l'état de l'art de la réduction formelle des systèmes linéaires singuliers d'équations différentielles ordinaires univariées (ODS). Dans le cas des systèmes différentiels linéaires singulièrement perturbés, les complications surviennent essentiellement à cause du phénomène des points tournants. Nous généralisons les notions et les algorithmes introduits pour le traitement des ODS afin de construire des solutions formelles. Les algorithmes sous-jacents sont également autonomes (par exemple la réduction de rang, la classification de la singularité, le calcul de l'indice de restriction). Dans le cas des systèmes de Pfaff, les complications proviennent de l'interdépendance des multiples sous-systèmes et de leur nature multivariée. Néanmoins, nous montrons que les invariants formels de ces systèmes peuvent être récupérés à partir d'un ODS associé, ce qui limite donc le calcul à des corps univariés. De plus, nous donnons un algorithme de réduction de rang et nous discutons des obstacles rencontrés. Outre ces deux systèmes, nous parlons des singularités apparentes des systèmes différentiels univariés dont les coefficients sont des fonctions rationnelles et du problème des valeurs propres perturbées. Les techniques développées au sein de cette thèse facilitent les généralisations d'autres algorithmes disponibles pour les systèmes différentiels univariés aux cas des systèmes bivariés ou multivariés, et aussi aux systèmes d''equations fonctionnelles
In this thesis, we are interested in the local analysis of singularly-perturbed linear differential systems and completely integrable Pfaffian systems in several variables. Such systems have a vast literature and arise profoundly in applications. However, their symbolic resolution is still open to investigation. Our approaches rely on the state of art of formal reduction of singular linear systems of ordinary differential equations (ODS) over univariate fields. In the case of singularly-perturbed linear differential systems, the complications arise mainly from the phenomenon of turning points. We extend notions introduced for the treatment of ODS to such systems and generalize corresponding algorithms to construct formal solutions in a neighborhood of a singularity. The underlying components of the formal reduction proposed are stand-alone algorithms as well and serve different purposes (e.g. rank reduction, classification of singularities, computing restraining index). In the case of Pfaffian systems, the complications arise from the interdependence of the multiple components which constitute the former and the multivariate nature of the field within which reduction occurs. However, we show that the formal invariants of such systems can be retrieved from an associated ODS, which limits computations to univariate fields. Furthermore, we complement our work with a rank reduction algorithm and discuss the obstacles encountered. The techniques developed herein paves the way for further generalizations of algorithms available for univariate differential systems to bivariate and multivariate ones, for different types of systems of functional equations

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14

Kunert, Gerd. "Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes." Universitätsbibliothek Chemnitz, 2000. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200000867.

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We consider a singularly perturbed reaction-diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.

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15

Kunert, Gerd. "Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100011.

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Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

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16

Kunert, Gerd. "A posteriori H^1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100730.

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The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

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17

Kunert, Gerd. "A posteriori error estimation for convection dominated problems on anisotropic meshes." Universitätsbibliothek Chemnitz, 2002. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200200255.

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A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.

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18

Kunert, Gerd. "A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes." Doctoral thesis, [S.l. : s.n.], 1999. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10324701.

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19

Hulek, Charlotte. "Systèmes d'équations différentielles linéaires singulièrement perturbées et développements asymptotiques combinés." Phd thesis, Université de Strasbourg, 2014. http://tel.archives-ouvertes.fr/tel-01021178.

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Dans ce travail nous démontrons un théorème de simplification uniforme concernant les équations différentielles ordinaires du second ordre singulièrement perturbées au voisinage d'un point dégénéré, appelé point tournant. Il s'agit d'une version analytique d'un résultat formel dû à Hanson et Russell, qui généralise un théorème connu de Sibuya. Pour traiter ce problème, nous utilisons les développements asymptotiques combinés Gevrey introduits par Fruchard et Schäfke. Dans une première partie nous rappelons les définitions et théorèmes principaux de cette récente théorie. Nous établissons trois résultats généraux que nous utilisons ensuite dans la seconde partie de ce manuscrit pour démontrer le théorème principal de réduction analytique annoncé. Enfin nous considérons des équations différentielles ordinaires d'ordre supérieur à deux, singulièrement perturbées à point tournant, et nous démontrons un théorème de réduction analytique.

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Musolino, Paolo. "Singular perturbation and homogenization problems in a periodically perforated domain. A functional analytic approach." Doctoral thesis, Università degli studi di Padova, 2012. http://hdl.handle.net/11577/3422452.

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This Dissertation is devoted to the singular perturbation and homogenization analysis of boundary value problems in the periodically perforated Euclidean space. We investigate the behaviour of the solutions of boundary value problems for the Laplace, the Poisson, and the Helmholtz equations, as parameters related to diameter of the holes or the size of the periodicity cells tend to 0. The Dissertation is organized as follows. In Chapter 1, we present two known constructions of a periodic analogue of the fundamental solution of the Laplace equation and we introduce the periodic layer and volume potentials for the Laplace equation and some basic results of periodic potential theory. Chapter 2 is devoted to singular perturbation and homogenization problems for the Laplace and the Poisson equations with Dirichlet and Neumann boundary conditions. In Chapter 3 we consider the case of (linear and nonlinear) Robin boundary value problems for the Laplace equation, while in Chapter 4 we analyze (linear and nonlinear) transmission problems. In Chapter 5 we apply the results of Chapter 4 in order to prove the real analyticity of the effective conductivity of a periodic dilute composite. Chapter 6 is dedicated to the construction of a periodic analogue of the fundamental solution of the Helmholtz equation and of the corresponding periodic layer potentials. In Chapter 7 we collect some results of spectral theory for the Laplace operator in periodically perforated domains. In Chapter 8 we investigate singular perturbation and homogenization problems for the Helmholtz equation with Neumann boundary conditions. In Chapter 9 we consider singular perturbation and homogenization problems with Dirichlet boundary conditions for the Helmholtz equation, while in Chapter 10 we study (linear and nonlinear) Robin boundary value problems. Chapter 11 is devoted to the study of periodic layer potentials for general second order differential operators with constant coefficients. At the end of the Dissertation we have enclosed some Appendices with some results that we have exploited.
Questa Tesi è dedicata all'analisi di problemi di perturbazione singolare e omogeneizzazione nello spazio Euclideo periodicamente perforato. Studiamo il comportamento delle soluzioni di problemi al contorno per le equazioni di Laplace, di Poisson e di Helmholtz al tendere a 0 di parametri legati al diametro dei buchi o alla dimensione delle celle di periodicità. La Tesi è organizzata come segue. Nel Capitolo 1, presentiamo due costruzioni note di un analogo periodico della soluzione fondamentale dell'equazione di Laplace, e introduciamo potenziali di strato e di volume periodici per l'equazione di Laplace e alcuni risultati basilari di teoria del potenziale periodica. Il Capitolo 2 è dedicato a problemi di perturbazione singolare e omogeneizzazione per le equazioni di Laplace e Poisson con condizioni al bordo di Dirichlet e Neumann. Nel Capitolo 3 consideriamo il caso di problemi al contorno di Robin (lineari e nonlineari) per l'equazione di Laplace, mentre nel Capitolo 4 analizziamo problemi di trasmissione (lineari e nonlineari). Nel Capitolo 5 applichiamo i risultati del Capitolo 4 al fine di provare l'analiticità della conduttività effettiva di un composto periodico. Il Capitolo 6 è dedicato alla costruzione di un analogo periodico della soluzione fondamentale dell'equazione di Helmholtz e dei corrispondenti potenziali di strato. Nel Capitolo 7 raccogliamo alcuni risultati di teoria spettrale per l'operatore di Laplace in domini periodicamente perforati. Nel Capitolo 8 studiamo problemi di perturbazione singolare e di omogeneizzazione per l'equazione di Helmholtz con condizioni al contorno di Neumann. Nel Capitolo 9 consideriamo problemi di perturbazione singolare e di omogeneizzazione con condizioni al contorno di Dirichlet per l'equazione di Helmholtz, mentre nel Capitolo 10 studiamo problemi al contorno di Robin (lineari e nonlineari). Il Capitolo 11 è dedicato allo studio di potenziali di strato periodici per operatori differenziali generali del secondo ordine a coefficienti costanti. Alla fine della Tesi abbiamo incluso delle Appendici con alcuni risultati utilizzati.

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21

Grosman, Sergey. "Adaptivity in anisotropic finite element calculations." Doctoral thesis, Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600815.

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When the finite element method is used to solve boundary value problems, the corresponding finite element mesh is appropriate if it is reflects the behavior of the true solution. A posteriori error estimators are suited to construct adequate meshes. They are useful to measure the quality of an approximate solution and to design adaptive solution algorithms. Singularly perturbed problems yield in general solutions with anisotropic features, e.g. strong boundary or interior layers. For such problems it is useful to use anisotropic meshes in order to reach maximal order of convergence. Moreover, the quality of the numerical solution rests on the robustness of the a posteriori error estimation with respect to both the anisotropy of the mesh and the perturbation parameters. There exist different possibilities to measure the a posteriori error in the energy norm for the singularly perturbed reaction-diffusion equation. One of them is the equilibrated residual method which is known to be robust as long as one solves auxiliary local Neumann problems exactly on each element. We provide a basis for an approximate solution of the aforementioned auxiliary problem and show that this approximation does not affect the quality of the error estimation. Another approach that we develope for the a posteriori error estimation is the hierarchical error estimator. The robustness proof for this estimator involves some stages including the strengthened Cauchy-Schwarz inequality and the error reduction property for the chosen space enrichment. In the rest of the work we deal with adaptive algorithms. We provide an overview of the existing methods for the isotropic meshes and then generalize the ideas for the anisotropic case. For the resulting algorithm the error reduction estimates are proven for the Poisson equation and for the singularly perturbed reaction-difussion equation. The convergence for the Poisson equation is also shown. Numerical experiments for the equilibrated residual method, for the hierarchical error estimator and for the adaptive algorithm confirm the theory. The adaptive algorithm shows its potential by creating the anisotropic mesh for the problem with the boundary layer starting with a very coarse isotropic mesh.

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22

Caillerie, Nils. "Équations cinétiques stochastiques et déterministes dans le contexte des mathématiques appliquées à la biologie." Thesis, Lyon, 2017. http://www.theses.fr/2017LYSE1117/document.

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Cette thèse étudie des modèles mathématiques inspirés par la biologie. Plus précisément, nous nous concentrons sur des équations aux dérivées partielles cinétiques. Les champs d'application des équations cinétiques sont nombreux mais nous nous concentrons ici sur des phénomènes de propagation d'espèces invasives, notamment la bactérie Escherichia coli et le crapaud buffle Rhinella marina.La première partie de la thèse ne présente pas de résultats mathématiques. Nous construisons plusieurs modélisations pour la dispersion à grande échelle du crapaud buffle en Australie. Nous confrontons ces mêmes modèles à des données statistiques multiples (taux de fécondité, taux de survie, comportements dispersifs) pour mesurer leur pertinence. Ces modèles font intervenir des processus à sauts de vitesses et des équations cinétiques.Dans la seconde partie, nous étudions des phénomènes de propagation dans des modèles cinétiques plus simples. Nous illustrons plusieurs méthodes pour établir mathématiquement des formules de vitesse de propagation dans ces modèles. Cette partie nous amène à établir des résultats de convergence d'équations cinétiques vers des équations de Hamilton-Jacobi par la méthode de la fonction test perturbée. Nous montrons également comment le formalisme Hamilton-Jacobi permet de trouver des résultats de propagation et enfin, nous construisons des solutions en ondes progressives pour un modèle de transport-réaction. Dans la dernière partie, nous établissons un résultat de limite de diffusion stochastique pour une équation cinétique aléatoire. Pour ce faire, nous adaptons la méthode de la fonction test perturbée sur la formulation d'une EDP stochastique en terme de générateurs infinitésimaux.La thèse comporte également une annexe qui expose les données trajectorielles des crapauds dont nous nous servons en première partie."
In this thesis, we study some biology inspired mathematical models. More precisely, we focus on kinetic partial differential equations. The fields of application of such equations are numerous but we focus here on propagation phenomena for invasive species, the Escherichia coli bacterium and the cane toad Rhinella marina, for example. The first part of this this does not establish any mathematical result. We build several models for the dispersion of the cane toad in Australia. We confront those very models to multiple statistical data (birth rate, survival rate, dispersal behaviors) to test their validity. Those models are based on velocity-jump processes and kinetic equations. In the second part, we study propagation phenomena on simpler kinetic models. We illustrate several methods to mathematically establish propagation speed in this models. This part leads us to establish convergence results of kinetic equations to Hamilton-Jacobi equations by the perturbed test function method. We also show how to use the Hamilton-Jacobi framework to establish spreading results et finally, we build travelling wave solutions for reaction-transport model. In the last part, we establish a stochastic diffusion limit result for a kinetic equation with a random term. To do so, we adapt the perturbed test function method on the formulation of a stochastic PDE in term of infinitesimal generators. The thesis also contains an annex which presents the data on toads’ trajectories used in the first part."

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23

Shakti, D. "Numerics of singularly perturbed differential equations." Thesis, 2014. http://ethesis.nitrkl.ac.in/6258/1/E-99.pdf.

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The main purpose of this report is to carry out the effect of the various numerical methods for solving singular perturbation problems on non-uniform meshes. When a small parameter epsilon known as the singular perturbation parameter is multiplied with the higher order terms of the differential equation, then the differential equation becomes singularly perturbed. In this type of problems, there are regions where the solution varies very rapidly known as boundary layers and the region where the solution varies uniformly known as the outer region. Standard finite difference/element methods are applied on the singularly perturbed differential equation on uniform mesh give unsatisfactory result as epsilon tends to zero. Due to presence of boundary layer, standard difference schemes unable to capture the layer behaviour until the mesh parameter and perturbation parameter are of the same size which results vast computational cost. In order to overcome this difficulty, we adapt non-uniform meshes. The Shishkin mesh and the adaptive mesh are two widely used special type of non-uniform meshes for solving singularly perturbed problem. Here, in this report singularly perturbed problems namely convection-diffusion and reaction-diffusion problems are considered and solved by various numerical techniques. The numerical solution of the problems are compared with the exact solution and the results are shown in the shape of tables and graphs to validate the theoretical bounds.

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24

Yadav, Sangeeta. "Data Driven Stabilization Schemes for Singularly Perturbed Differential Equations." Thesis, 2023. https://etd.iisc.ac.in/handle/2005/6095.

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This thesis presents a novel way of leveraging Artificial Neural Network (ANN) to aid conventional numerical techniques for solving Singularly Perturbed Differential Equation (SPDE). SPDEs are challenging to solve with conventional numerical techniques such as Finite Element Methods (FEM) due to the presence of boundary and interior layers. Often the standard numerical solution shows spurious oscillations in the vicinity of these layers. Stabilization techniques are often employed to eliminate these spurious oscillations in the numerical solution. The accuracy of the stabilization technique depends on a user-chosen stabilization parameter whose optimal value is challenging to find. A few formulas for the stabilization parameter exist in the literature, but none extends well for high-dimensional and complex problems. In order to solve this challenge, we have developed the following ANN-based techniques for predicting this stabilization parameter: 1) SPDE-Net: As a proof of concept, we have developed an ANN called SPDE-Net for one-dimensional SPDEs. In the proposed method, we predict the stabilization parameter for the Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. The prediction task is modelled as a regression problem using equation coefficients and domain parameters as inputs to the neural network. Three training strategies have been proposed, i.e. supervised learning, L 2-Error minimization (global) and L2-Error minimization (local). The proposed method outperforms existing state-of-the-art ANN-based partial differential equations (PDE) solvers, such as Physics Informed Neural Networks (PINNs). 2) AI-stab FEM With an aim for extending SPDE-Net for two-dimensional problems, we have also developed an optimization scheme using another Neural Network called AI-stab FEM and showed its utility in solving higher-dimensional problems. Unlike SPDE-Net, it minimizes the equation residual along with the crosswind derivative term and can be classified as an unsupervised method. We have shown that the proposed approach yields stable solutions for several two-dimensional benchmark problems while being more accurate than other contemporary ANN-based PDE solvers such as PINNs and Variational Neural Networks for the Solution of Partial Differential Equations (VarNet) 3) SPDE-ConvNet In the last phase of the thesis, we attempt to predict a cell-wise stabilization parameter to treat the interior/boundary layer regions adequately by developing an oscillations-aware neural network. We present SPDE-ConvNet, Convolutional Neural Network (CNN), for predicting the local (cell-wise) stabilization parameter. For the network training, we feed the gradient of the Galerkin solution, which is an indirect metric for representing oscillations in the numerical solution, along with the equation coefficients, to the network. It obtains a cell-wise stabilization parameter while sharing the network parameters among all the cells for an equation. Similar to AI-stab FEM, this technique outperforms PINNs and VarNet. We conclude the thesis with suggestions for future work that can leverage our current understanding of data-driven stabilization schemes for SPDEs to develop and improve the next-generation neural network-based numerical solvers for SPDEs.

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25

Goswami, Amartya. "Asymptotic analysis of singularly perturbed dynamical systems." Thesis, 2011. http://hdl.handle.net/10413/9760.

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According to the needs, real systems can be modeled at various level of resolution. It can be detailed interactions at the individual level (or at microscopic level) or a sample of the system (or at mesoscopic level) and also by averaging over mesoscopic (structural) states; that is, at the level of interactions between subsystems of the original system (or at macroscopic level). With the microscopic study one can get a detailed information of the interaction but at a cost of heavy computational work. Also sometimes such a detailed information is redundant. On the other hand, macroscopic analysis, computationally less involved and easy to verify by experiments. But the results obtained may be too crude for some applications. Thus, the mesoscopic level of analysis has been quite popular in recent years for studying real systems. Here we will focus on structured population models where we can observe various level of organization such as individual, a group of population, or a community. Due to fast movement of the individual compare of the other demographic processes (like death and birth), the problem is multiple-scale. There are various methods to handle multiple-scale problem. In this work we will follow asymptotic analysis ( or more precisely compressed Chapman–Enskog method) to approximate the microscopic model by the averaged one at a given level of accuracy. We also generalize our model by introducing reducible migration structure. Along with this, considering age dependency of the migration rates and the mortality rates, the thesis o ers improvement of the existing literature.
Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.

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26

Li, Fang-wen, and 李芳雯. "Radial Basis Collocation Method for Singularly Perturbed Partial Differential Equations." Thesis, 2004. http://ndltd.ncl.edu.tw/handle/06442167851802632152.

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碩士
國立中山大學
應用數學系研究所
92
In this thesis, we integrate the particular solutions of singularly perturbed partial differential equations into radial basis collocation method to solve two kinds of boundary layer problem.

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27

Govindarao, Lolugu. "Parameter Uniform Numerical Methods for Singularly Perturbed Parabolic Partial Differential Equations." Thesis, 2019. http://ethesis.nitrkl.ac.in/10169/1/2019_PhD_LGovindarao_515MA6012_Parameter.pdf.

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This thesis deals with some efficient and higher order numerical methods for solving singularly perturbed parabolic partial differential equations (SPPDEs) in one and two dimensions. The model problems includes one dimensional SPPDEs, time delay SPPDEs, two dimensional SPPDEs and mixed type of parabolic-elliptic problems. In general, these problems are described by partial differential equations in which the highest order derivative is multiplied by a small parameter ε, known as the “singular perturbation parameter” (0 < ε ≪ 1). If the parameter ε tends to 0, the problem has a limiting solution, which is called the solution of the reduced problem. The regions of nonuniform convergence lie near the boundary of the domain, which are known as boundary/interior layers. Due to this layer phenomena, it is a very difficult and challenging task to provide parameter uniform numerical methods for solving SPPDEs. The term “parameter uniform” is meant to identify those numerical methods, in which the approximate solution converges (measured in the supremum norm) independently with respect to the perturbation parameter. The purpose of the thesis is to analyze, develop and optimize the parameter uniform fitted mesh methods for solving SPPDEs on Shishkin-type meshes like the standard Shishkin mesh (S-mesh), the Bakhvalov-Shishkin mesh (B-S mesh), and the modified Bakhvalov-Shishkin mesh (M-B-S mesh) in the spacial direction. This thesis contains eight chapters. It begins with introduction along with the objective and the motivation for solving SPPDEs. Next, Chapter 2 contains a time delay SPPDE which is solved using a hybrid scheme (combination of the midpoint upwind scheme and the central difference scheme) on S-mesh and B-S mesh in space direction and the implicit trapezoidal scheme on uniform mesh in time direction. We have obtained an optimal global second order accuracy with respect to space and time. In Chapter 3, a monotone hybrid scheme for discretization in space and the implicit Euler, then the implicit trapezoidal scheme for discretization in time are used to solve a SPPDE. A monotone hybrid scheme is different from the usual hybrid scheme, which is a combination of the midpoint and the central difference scheme with variable weights. Here, an optimal second order accuracy is obtained in space and time. Again, the same monotone hybrid scheme in spatial direction and the Euler scheme in time direction are used in Chapter 4 for solving the model problem considered in Chapter 2. Chapter 5 contains a singularly perturbed time delay reaction-diffusion problem and it is solved by using the central difference scheme on S-mesh and B-S mesh in space direction and the implicit Euler, the implicit trapezoidal scheme in time direction to get a second order accuracy. Then, the post-processing technique (Richardson extrapolation) is used to improve the accuracy from second order to fourth order. Chapter 6 presents the hybrid scheme on Shishkin-type meshes in spatial direction and the implicit Euler, the Crack-Nicolson schemes in time direction for solving the singularly perturbed mixed type of parabolic-elliptic problem. Next, we extend from 1D problem to 2D problem SPPDE in Chapter 7, in which a 2D SPPDE is solved by using the central difference scheme on Shishkin-type meshes in space direction and the Peaceman-Rachford scheme in time direction. Here, we have achieved an optimal second order accuracy in both spatial and temporal direction. In all cases, Thomas algorithm is used throughout this thesis to reduce the computational time over the usual matrix inverse method. Finally, Chapter 8 summarizes the results made by this thesis. Extensive numerical results are presented in support of the theoretical findings and also to demonstrate the accuracy of the proposed methods. The corresponding numerical results are presented in the numerical section of each chapter of the thesis in shape of tables and figures. Some comparison results are also provided which confirms the efficiency of the proposed methods made in this thesis over the existing methods in literature.

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28

"A higher-order energy expansion to two-dimensional singularly perturbed Neumann problems." 2004. http://library.cuhk.edu.hk/record=b5891877.

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Анотація:

Yeung Wai Kong.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.
Includes bibliographical references (leaves 51-55).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.5
Chapter 2 --- Some Preliminaries --- p.13
Chapter 3 --- "Approximate Function we,p" --- p.17
Chapter 4 --- "The Computation Of Je[we,p]" --- p.21
Chapter 5 --- The Signs of c1 And c3 --- p.30
Chapter 6 --- The Asymptotic Behavior of ue and Je[ue] --- p.35
Chapter 7 --- "The Proofs Of Theorem 1.1, Theorem 1.2 And Corol- lary 11" --- p.40
Appendix --- p.43
Bibliography --- p.51

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29

Reibiger, Christian. "Optimal Control Problems with Singularly Perturbed Differential Equations as Side Constraints: Analysis and Numerics." Doctoral thesis, 2014. https://tud.qucosa.de/id/qucosa%3A28578.

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It is well-known that the solution of a so-called singularly perturbed differential equation exhibits layers. These are small regions in the domain where the solution changes drastically. These layers deteriorate the convergence of standard numerical algorithms, such as the finite element method on a uniform mesh. In the past many approaches were developed to overcome this difficulty. In this context it was very helpful to understand the structure of the solution - especially to know where the layers can occur. Therefore, we have a lot of analysis in the literature concerning the properties of solutions of such problems. Nevertheless, this field is far from being understood conclusively. More recently, there is an increasing interest in the numerics of optimal control problems subject to a singularly perturbed convection-diffusion equation and box constraints for the control. However, it is not much known about the solutions of such optimal control problems. The proposed solution methods are based on the experience one has from scalar singularly perturbed differential equations, but so far, the analysis presented does not use the structure of the solution and in fact, the provided bounds are rather meaningless for solutions which exhibit boundary layers, since these bounds scale like epsilon^(-1.5) as epsilon converges to 0. In this thesis we strive to prove bounds for the solution and its derivatives of the optimal control problem. These bounds show that there is an additional layer that is weaker than the layers one expects knowing the results for scalar differential equation problems, but that weak layer deteriorates the convergence of the proposed methods. In Chapter 1 and 2 we discuss the optimal control problem for the one-dimensional case. We consider the case without control constraints and the case with control constraints separately. For the case without control constraints we develop a method to prove bounds for arbitrary derivatives of the solution, given the data is smooth enough. For the latter case we prove bounds for the derivatives up to the second order. Subsequently, we discuss several discretization methods. In this context we use special Shishkin meshes. These meshes are piecewise equidistant, but have a very fine subdivision in the region of the layers. Additionally, we consider different ways of discretizing the control constraints. The first one enforces the compliance of the constraints everywhere and the other one enforces it only in the mesh nodes. For each proposed algorithm we prove convergence estimates that are independent of the parameter epsilon. Hence, they are meaningful even for small values of epsilon. As a next step we turn to the two-dimensional case. To be able to adapt the proofs of Chapter 2 to this case we require bounds for the solution of the scalar differential equation problem for a right hand side f only in W^(1,infty). Although, a lot of results for this problem can be found in the literature but we can not apply any of them, because they require a smooth right hand side f in C^(2,alpha) for some alpha in (0,1). Therefore, we dedicate Chapter 3 to the analysis of the scalar differential equations problem only using a right hand side f that is not very smooth. In Chapter 4 we strive to prove bounds for the solution of the optimal control problem in the two dimensional case. The analysis for this problem is not complete. Especially, the characteristic layers induce subproblems that are not understood completely. Hence, we can not prove sharp bounds for all terms in the solution decomposition we construct. Nevertheless, we propose a solution method. Numerical results indicate an epsilon-independent convergence for the considered examples - although we are not able to prove this.

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30

COLUZZI, BARTOCCIONI BARBARA. "Theoretical models and numerical methods for the study of sub-cellular phenomena." Doctoral thesis, 2018. http://hdl.handle.net/11573/1186551.

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Nella presente tesi si discute sia la transizione di denaturazione del DNA che la dinamica enzimatica di Michaelis-Menten, introducendo entrambi gli argomenti partendo dalla loro importanza dal punto di vista di una migliore comprensione dei fenomeni intra-cellulari. Vengono quindi presentati i risultati originali ottenuti. Si è effettuata un'analisi approfondita di dati numerici su un modello disordinato di Poland-Scheraga per la transizione di denaturazione del DNA in cui l'effetto di auto-evitamento è tenuto correttamente in considerazione, nella quale: i) sono state introdotte delle appropriate pseudo-temperature critiche dipendenti dalla sequenza, il che ha permesso intanto di ottenere una stima rifinita dell'esponente che caratterizza il comportamento al punto critico disordinato, in accordo con una transizione di fase di ordine maggiore del secondo; ii) sulla base di questa analisi si è inoltre potuto caratterizzare il lento approccio all'equilibrio termodinamico osservato introducendo un'appropriata lunghezza di crossover, definita come la lunghezza delle sequenze al di sopra della quale l'effetto del disordine diviene evidente (sia dal comportamento delle varie osservabili mediate sulle sequenze, sia da quello in particolare del parametro d'ordine e della suscettività in circa la metà delle singole sequenze); iii) infine, si è descritto in dettaglio uno scenario fenomenologico nell'ambito del quale la lunghezza di crossover viene messa in relazione con i parametri del modello, e quindi, attraverso il calcolo combinatoriale della probabilità di ottenere una regione rara di lunghezza L in una sequenza di lunghezza N, si possono ottenere delle predizioni sul comportamento di taglia finita per diversi valori dei parametri. Nel caso della dinamica enzimatica di Michaelis Menten, si è portato a termine un dettagliato studio analitico, partendo dall'approssimazione standard di stato quasi-stazionario, che chiarisce le similitudini e le differenze tra l'approccio alternativo che si è introdotto, basato su tecniche di gruppo di rinormalizzazione, ed il metodo perturbativo che si è soliti applicare a sistemi ad effetto strato come quello considerato: i) in particolare, si è arrivati al secondo ordine nello sviluppo nel parametro appropriato, ottenendo corrispondentemente per la prima volta le soluzioni interne a quest'ordine, che non erano note in letteratura; ii) sulla base dell'analisi del comportamento delle approssimazioni uniformi così ottenute, alcune delle cui caratteristiche appaiono iterabili, si è potuto predire anche una parte del contributo a quest'ordine delle soluzioni esterne, quindi delle approssimazioni uniformi che riproducono il comportamento numerico delle soluzioni meglio di quelle note in una larga parte della finestra di tempo in cui svolge il fenomeno, anche in un caso studiato particolarmente sfavorevole sia dal punto di vista dei valori delle costanti cinetiche che di quello del parametro di espansione, tendendo inoltre correttamente a zero asintoticamente; iii) il metodo introdotto risulta quindi efficace, e le verifiche che sono state fatte dovrebbero permettere la sua futura applicazione intanto alla dinamica enzimatica di Michaelis Menten nell'ambito dell'approssimazione totale di stato quasi-stazionario.

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31

Park, Peter J. "Multiscale numerical methods for the singularly perturbed convection-diffusion equation." Thesis, 2000. https://thesis.library.caltech.edu/783/1/Park_pj_2000.pdf.

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We develop efficient and robust numerical methods in the finite element framework for numerical solutions of the singularly perturbed convection-diffusion equation and of a degenerate elliptic equation. The standard methods for purely elliptic or hyperbolic problems perform poorly when there are sharp boundary and internal layers in the solution caused by the dominant convective effect. We offer a new approach in which we design the finite element basis functions that capture the local behavior correctly. When the structure of the layers can be determined locally, we apply the multiscale finite element method in which we solve the corresponding homogeneous equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. When the layer structure is nonlocal, we use a variational principle to gain additional information. For a random velocity field, this variational principle provides correct scaling results. This allows us to design asymptotic basis functions that can capture the global layers correctly. The same approach is also extended to elliptic problems with high contrast coefficients. When an asymptotic result is available, it is incorporated naturally into the finite element setting developed earlier. When there is a strong singularity due to a discontinuous coefficient, we construct the basis functions using the infinite element method. Our methods can handle singularities efficiently and are not sensitive to the large contrast.

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32

Lamač, Jan. "Adaptivní metody pro singulárně porušené parciální diferenciální rovnice." Doctoral thesis, 2017. http://www.nusl.cz/ntk/nusl-368918.

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This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.

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