The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.     

An accurate semi-analytic finite difference scheme for two-dimensional elliptic problems with singularities

A high-order semi-analytic finite difference scheme is presented to overcome degradation of numerical performance when applied to two-dimensional elliptic problems containing singular points. The scheme, called Least-Square Singular Finite Difference Scheme (L-S SFDS), applies an explicit functional representation of the exact solution in the vicinity of the singularities, and a conventional finite difference scheme on the remaining domain. It is shown that the L-S SFDS is “pollution” free, i.e., no degradation in the convergence rate occurs because of the singularities, and the coefficients of the asymptotic solution in the vicinity of the singularities are computed as a by-product with a very high accuracy. Numerical examples for the Laplace and Poisson equations over domains containing re-entrant corners or abrupt changes in the boundary conditions are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 281–296, 1998     

A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations

A new explicit formula for the integrals of shifted Chebyshev polynomials of any degree for any fractional-order in terms of shifted Chebyshev polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of linear multi-order fractional differential equations (FDEs) by considering their integrated forms. The shifted Chebyshev spectral tau (SCT) method based on the integrals of shifted Chebyshev polynomials is applied to construct the numerical solution for such problems. The method is then tested on examples. It is shown that the SCT yields better results.     

Convergence theorems for HL integrable vector-valued functions with applications

In this paper we prove dominated and monotone convergence theorems for HL integrable Banach-valued functions. These results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of Fredholm type in ordered Banach spaces involving Kurzweil integrals or improper integrals. Results are used also to solve concrete second-order functional boundary value problems involving discontinuities and singularities.     

A DIRECT SEARCH FRAME-BASED CONJUGATE GRADIENTS METHOD

A derivative-free frame-based conjugate gradients algorithm is presented.Convergenceis shown for C~1 functions,and this is verified in numerical trials.The algorithm is tested ona variety of low dimensional problems,some of which are ill-conditioned,and is also testedon problems of high dimension.Numerical results show that the algorithm is effectiveon both classes of problems.The results are compared with those from a discrete quasi-Newton method,showing that the conjugate gradients algorithm is competitive.Thealgorithm exhibits the conjugate gradients speed-up on problems for which the Hessian atthe solution has repeated or clustered eigenvalues.The algorithm is easily parallelizable.     

Problems and methods in partial differential equations

Summary The method of singularities is used to solve theCauchy problem for simple hyperbolic partial differential equations, namely, the wave equation and the damped wave equation. The representation formula for the solution of theCauchy problem is written in terms of finite parts and logarithmic parts of certain divergent integrals. A process of analytic continuation is also used to solve theCauchy problems under consideration. However, to obtain explicitly the representation formulas for the solutions, one must actually perform the analytic continuation. It is shown that this is best achieved by making use of finite and logarithmic parts. Simple examples were purposely chosen so as to show that consideration of finite and logarithmic parts is naturally unavoidable and ? in the very nature of things ?. To Enrico Bompiani on his scientific Jubilee. This work was sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States Air Force, through its European Office.     

Quadpack computation of Feynman loop integrals

The paper addresses a numerical computation of Feynman loop integrals, which are computed by an extrapolation to the limit as a parameter in the integrand tends to zero. An important objective is to achieve an automatic computation which is effective for a wide range of instances. Singular or near singular integrand behavior is handled via an adaptive partitioning of the domain, implemented in an iterated/repeated multivariate integration method. Integrand singularities possibly introduced via infrared (IR) divergence at the boundaries of the integration domain are addressed using a version of the Dqags algorithm from the integration package Quadpack, which uses an adaptive strategy combined with extrapolation. The latter is justified for a large class of problems by the underlying asymptotic expansions of the integration error. For IR divergent problems, an extrapolation scheme is presented based on dimensional regularization.     

Formation of singularities of solutions of the equations of motion of compressible fluids subjected to external forces in the case of several spatial variables

One considers a class of solutions with finite total energy and moment of inertia for the equations of motion of compressible fluids. It is shown that for a wide class of right-hand sides, including the viscosity term, initially smooth solutions may acquire singularities on a finite time interval. A sufficient condition for the appearance of singularities is found. This condition may be called “the best possible sufficient condition” in the sense that one can explicitly construct a time-global smooth solution for which this condition does not hold to within arbitrary infinitely small quantities. For a nontrivial constant state, perturbations with compact support are considered. A generalization is proved for the known theorem on the initial conditions for which the solution acquires singularities on a finite time interval. The effect of dry friction and rotation on the formation of singularities of smooth solutions is examined. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 274–308, 2007.     

Sampling with a String

Kramer's sampling theorem forms a bridge between the Whittaker-Shannon-Kotel'nikov sampling theorem and boundary-value problems. It has been shown that sampling expansions associated with Sturm-Liouville boundary-value problems are Lagrange-type sampling series, i.e., Lagrange series with infinitely many terms converging to entire functions. String theory as developed by Feller, Kac, and Krein, is a generalization of the Sturm-Liouville theory. We investigate sampling series associated with strings and compare them with those associated with Sturm-Liouville problems. We show that unlike sampling series associated with Sturm-Liouville problems, those associated with strings include not only Lagrange-type sampling series, but also Lagrange polynomial interpolation.     

A new way to evaluate the Probability and Fresnel integrals

In this article, we show how Laplace Transform may be used to evaluate variety of nontrivial improper integrals, including Probability and Fresnel integrals. The algorithm we have developed here to evaluate Probability, Fresnel and other similar integrals seems to be new. This method transforms the evaluation of certain improper integrals into evaluation of improper integrals of the corresponding Laplace transform, which in many cases are much easier.     

A new Tikhonov regularization method

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size, which allow evaluation of the singular value decomposition of the matrix defining the problem, are the truncated singular value decomposition and Tikhonov regularization. The present paper proposes a novel choice of regularization matrix for Tikhonov regularization that bridges the gap between Tikhonov regularization and truncated singular value decomposition. Computed examples illustrate the benefit of the proposed method.     

Discretization of second-order ordinary differential equations with symmetries

A number of publications (indicated in the Introduction) are overviewed that address the group properties, first integrals, and integrability of difference equations and meshes approximating second-order ordinary differential equations with symmetries. A new example of such equations is discussed in the overview. Additionally, it is shown that the parametric families of invariant difference schemes include exact schemes, i.e., schemes whose general solution coincides with the corresponding solution set of the differential equations at mesh nodes, which can be of arbitrary density. Thereby, it is shown that there is a kind of mathematical dualism for the problems under study: for a given physical process, there are two mathematical models: continuous and discrete. The former is described by continuous curves, while the latter, by points on these curves.     

Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations

A new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The Bernstein–Petrov–Galerkin method (BPG) is applied to construct the numerical solution for such problems. The method is then tested on examples and compared with other methods. It is shown that the BPG yields better results.     

On integrals of equations of unstable nearly self-similar flows : PMM vol. 39, n 6, 1975, pp. 1060–1067

One-dimensional or nearly one-dimensional unstable motions of perfect gas are considered. Integrals admitted by the system of equations defining such motions are examined. Since the existence of integrals is associated with some law of conservation, i. e. with some divergent form of presentation of equations of the input system, it is possible by examining all divergent equations of gasdynamics to derive certain new integrals not previously considered.     

Inexact smoothing method for large scale minimax optimization

A new algorithm for the solution of large scale minimax problems of a finite number of functions is introduced. The algorithm is a smoothing method based on a maximum entropy function and an inexact Newton-type algorithm for its solution. Under mild assumptions, only the approximate solution of a linear system is required at each iteration. The algorithm is shown to both globally and superlinearly convergent. Meanwhile some implementation techniques taking advantage of the sparsity of the Hessians of the functions and alleviating the disadvantage effect of the ill-conditioned matrix are considered. Numerical results show that the inexact method is considerably efficient.     

Numerical diagnostics of solution blowup in differential equations

New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.