High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems. © 2022 by the authors.King Saud University, KSUM. A. Zaky and A. Aldraiweesh extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia)

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

Impulsive Hybrid Discrete-Continuous Delay Differential Equations

This thesis deals with impulsive hybrid discrete-continuous delay differential equations (IHDDEs). This new class of differential equations is highly challenging for two reasons. First, because of a dependency of the right-hand-side function on past states, with time delays that depend on the current state. Second, because both the right-hand-side function and the state itself are discontinuous at implicitly defined time points. The theoretical results and numerical methods presented in this thesis are related to the following subject areas: First, solutions of initial value problems (IVPs) in IHDDEs. Second, derivatives of IVP solutions with respect to parameters (“sensitivities”). Third, estimation of parameters in IHDDE models from experimental data. Amongst others, this thesis thereby makes the following contributions: - The theoretical basis of IHDDE-IVPs is established. This includes the definition of a solution concept, the existence of solutions, the uniqueness of solutions, and the differentiability of solutions with respect to parameters. - A new approach for numerically solving IVPs in differential equations with time delays is introduced. A key aspect is the use of extrapolations beyond past discontinuities. Convergence of continuous Runge-Kutta methods realized in the framework of the new approach is shown, and numerical results are presented that demonstrate the benefit of using extrapolations on a practical example. - A “first discretize, then differentiate” approach and a “first differentiate, then discretize” approach for forward sensitivity computation in IHDDEs are investigated. It is revealed that the presence of time delays destroys commutativity of differentiation and discretization in the case of continuous Runge-Kutta methods. - An extension of the concept of Internal Numerical Differentiation is proposed for differential equations with time delays. The use of the extended concept ensures that numerically computed sensitivities converge to the exact sensitivities, and that the convergence order is identical to the convergence order of the method that is used for solving the nominal IVP. - The first practical forward and adjoint schemes are developed that realize Internal Numerical Differentiation for IHDDEs. Numerical investigations show that the developed schemes are drastically more efficient than classical methods for sensitivity computation. - The new numerical methods for solving IVPs and for computing sensitivites are successfully applied to several challenging test cases, and the properties of the methods are analysed. - Numerical methods are presented for solving nonlinear least-squares parameter estimation problems constrained by IHDDEs. - A new epidemiological IHDDE model is developed. Therein, an impulse accounts for the arrival of an infected population. Further, the zeros of state-dependent switching functions characterize the time points at which new medical treatments become available. - A delay differential equation model is presented for the crosstalk of the signaling pathways of two cytokines. In comparison to an ordinary differential equation model, a better fit to experimental data is obtained with a smaller number of differential states. - A novel model is proposed to describe the voting behavior of the viewers of the TV singing competition “Unser Star für Baku” aired in 2012. Numerical results show that the use of a time delay is crucial for a qualitative correct description of the voting behavior. Furthermore, parameter estimation results yield a good quantitative agreeement with data from the TV show. - The practical implementation of all developed methods in the new software packages Colsol-DDE and ParamEDE is described

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described