Mathematics department, ADMC

We discuss a semidiscrete framework for nonlinear di usion scale-spaces, where the image is sampled on a nite grid and the scale parameter is continuous. This leads to a system of nonlinear ordinary di erential equations. We investigate conditions under which one can guarantee well-posedness properties, an extremum principle, average grey level invariance, smoothing Lyapunov functionals, and convergence to a constant steady-state. These properties are in analogy to previously established results for the continuous setting. Interestingly, this semidiscrete framework helps to explain the so-called Perona{Malik paradox: The Perona{Malik equation is a forward{ backward di usion equation which is widely-used in image processing since it combines intraregional smoothing with edge enhancement. Although its continuous formulation is regarded to be ill-posed, it turns out that a spatial discretization is su cient to create a well-posed semidiscrete di usion scale-space. We also prove that an explicit 1-D Perona{Malik scheme is monotonicity preserving, which explains that staircasing is essentially the only practically appearing instability.

Although the widely-used Perona{Malik lter is regarded as ill-posed, straightforward implementations are often surprisingly stable. We give an explanation for this e ect by applying a discrete nonlinear scale-space framework: a spatial discretization on a xed pixel grid gives a well-posed scale-space with many image-simplifying properties, and an explicit time discretization leads to a scheme which does not introduce additional oscillations. This explains why staircasing is essentially the only practically appearing instability.

This article proposes Laplace Transform Homotopy Perturbation Method (LT-HPM) to find an approximate solution for the problem of an axisymmetric Newtonian fluid squeezed between two large parallel plates. After comparing figures between approximate and exact solutions, we will see that the proposed solutions besides of handy, are highly accurate and therefore LT-HPM is extremely efficient.

This article proposes non-linearities distribution Laplace transform-homotopy perturbation method (NDLT-HPM) to find approximate solutions for linear and nonlinear differential equations with finite boundary conditions. We will see that the method is particularly relevant in case of equations with nonhomogeneous non-polynomial terms. Comparing figures between approximate and exact solutions we show the effectiveness of the proposed method.

This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant.

In this work, we present a technique for the analytical solution of systems of stiff ordinary differential equations (SODEs) using the power series method (PSM). Three SODEs systems are solved to show that PSM can find analytical solutions of SODEs systems in convergent series form. Additionally, we propose a post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a powerful technique to find exact solutions. The proposed method gives a simple procedure based on a few straightforward steps.

In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations.

The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient.

This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful technique to find exact solutions. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend on a perturbation parameter.

In this paper, we propose an efficient modification of a New Homotopy Perturbation Method (NHPM) to obtain approximate and exact analytical solutions of Partial Differential-Algebraic Equations (PDAEs). The NHPM is first applied to the PDAE to obtain the exact solution in convergent series form. To improve the solution obtained from NHPM's truncated series, a post-treatment combining Laplace transform and Padé approximant is proposed. This modified Laplace-Padé new homotopy perturbation method is shown to be effective and greatly improves NHPM's truncated series solutions in convergence rate, and often leads to the exact solution. Two problems are solved to demonstrate the efficiency of the method; the first one is a nonlinear index-one system with an integral term and the second one is a linear index-three system with variable coefficients.

Circuit simulation aids to predict and improve analog circuits performance. Direct current (DC) simulation highlights as a key tool to analyse linear and nonlinear circuits. Then, during the recent decades homotopy continuation methods (HCM) have been proposed as replacement for Newton-Raphson method (NR) to obtain operating points (OP) of nonlinear circuits. HCM has the ability to find multiple OP with global convergence, whilst NR method can only find one OP due to its characteristic of local convergence. Nonetheless, HCM lacks of a formal stop criterion to end DC simulations. Therefore, in this work, we propose a new double bounded homotopy (DBH) as a tool to find multiple OP of nonlinear circuits with a reliable mathematical stop criterion based on its property of forming closed trajectories. This kind of homotopy joins the other two reported DBH. Hence, we will show a performance comparison between the proposed method and the other two existent DBH; resulting in a better performance by the proposed HCM for a benchmark study case circuit having nine OP and containing bipolar transistors as nonlinear elements.

This paper proposes power series method (PSM) in order to find solutions for singular partial differential-algebraic equations (SPDAEs). We will solve three examples to show that PSM method can be used to search for analytical solutions of SPDAEs. What is more, we will see that, in some cases, Padé posttreatment, besides enlarging the domain of convergence, may be employed in order to get the exact solution from the truncated series solutions of PSM.

The slider-crank mechanism (SCM) is one of the most important mechanisms in modern technology. It appears in most combustion engines including those of automobiles, trucks, and other small engines. The SCM model considered here is an index-three nonlinear system of differential-algebraic equations (DAEs), and therefore difficult to integrate numerically. In this work, we present the application of the differential transform method (DTM) to obtain an approximate analytical solution of the SCM model in convergent series form. In addition, we propose a posttreatment of the power series solution with the Padé resummation method to extend the domain of convergence of the approximate series solution. The main advantage of the proposed technique is that it does not require an index reduction and does not generate secular terms or depend on a perturbation parameter.

This work presents the application of the power series method (PSM) to find solutions of nonlinear delay differential equations of pantograph type (PDDEs). Three equations are solved to show that PSM can provide analytical solutions of PDDEs in convergent series form. The nonlinear pantograph cases study are: a first order equation, a second order equation, and a second order singular equation. Additionally, we present the post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a powerful technique to find exact solutions. The proposed methodology possesses a simple procedure based on a few straightforward steps and it does not depend on a perturbation parameter.

In this paper, we discuss new advantages of using stratified databases of dynamic questions to construct customized exams automatically. Using this technique a large number of different random exams can be generated automatically without concerns about duplication or exposure of questions. This advantage is in particular important for large-lecture courses. We also show that a few updates to the questions database lead to a large increase in the number of exams. Furthermore, we derive an invulnerability criterion for exams and show that for a reasonably large database only few updates are required to maintain its security if the number of examinees increases. In this first attempt, the discussed technique was implemented using the Exam Builder of the Scientific WorkPlace to generate exams for the Pre-calculus and Calculus courses.

Rank-revealing ULV and URV factorizations are useful tools to determine the rank and to compute bases for null-spaces of a matrix. However, in the practical ULV (resp. URV) factorization each left (resp. right) null vector is recomputed from its corresponding right (resp. left) null vector via triangular solves. Triangular solves are required at initial factorization, re nement and updating. As a result, algorithms based on these factorizations may be expensive, especially on parallel computers where triangular solves are expensive. In this paper we propose an alternative approach. Our new rankrevealing ULV factorization, which we call "top-down" ULV factorization (TDULV-factorization) is based on right null vectors of lower triangular matrices and therefore no triangular solves are required. Right null vectors are easy to estimate accurately using condition estimators such as incremental condition estimator (ICE). The TDULV factorization is shown to be equivalent to the URV factorization with the advantage of circumventing triangular solves. Recently, Stewart 25, 26, 27, 29] proposed two rank-revealing factorizations, called ULV and URV factorizations. These two factorizations are

Constrained multibody mechanical systems are used in various applications. These systems often lead to index-2 or index-3 differential-algebraic equations (DAEs), which are known to pose a challenge to numerical integration methods. This paper develops a novel approach to solve index-2 DAEs which describe the dynamics of constrained multibody mechanical systems. The method, called PSMAP, relies on effective combination of the power series method (PSM) with Adomian polynomials (AP). To overcome the limitation of the PSM, the PSMAP expands nonlinear terms once in series form using AP. Further, by exploiting an important property of AP, a nonsingular linear algebraic system for the coefficients of the power series solution is derived and solved. To extend the domain of the power series solution, a multistage PSMAP is developed. The main advantage of our technique is that it applies directly to the DAE without the need for complex or costly transformations like index-reductions. Consequently, the PSMAP considerably reduces the computation work, leads to simple algorithm and avoids constraints violation. To demonstrate the efficiency of the PSMAP, two examples of two-link planar robotic systems are solved. Numerical results show that the PSMAP is a powerful tool to solve this class of DAEs.

En este artículo el Método de Perturbación (PM) es empleado para obtener una solución aproximada para el problema de Troesch. Además describiremos el uso de la Transformada de Laplace y la Aproximación de Padé para trabajar con las series truncadas obtenidas por el Método de Perturbación, y así obtener soluciones aproximadas compactas. Finalmente se propone una tabla comparativa entre la solución propuesta y otras soluciones reportadas en la literatura: Método de Descomposición de Adomian, Método de Perturbación Homotópica, Método de Análisis Homotópico y la solución numérica exacta. Los resultados muestran que nuestra solución es la más exacta (Error Relativo Absoluto Promedio ).

Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. e present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. is algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. is new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. is algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals.