Quadrature methods for integro-differential equations of Prandtl's type in weighted spaces of continuous functions

The paper deals with the approximate solution of integro-differential equations of Prandtl's type. Quadrature methods involving ``optimal'' Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved. The efficiency of the method has been tested by some numerical experiments, some of them including comparisons with other numerical procedures. In particular, as an application, we have implemented the method for solving Prandtl's equation governing the circulation air flow along the contour of a plane wing profile, in the case of elliptic or rectangular wing-shape.

[1]  Giovanni Monegato,et al.  Numerical evaluation of hypersingular integrals , 1994 .

[2]  C. Laurita.  Condition numbers for singular integral equations in weighted L 2 spaces , 2000 .

[3]  Elena Marchetti,et al.  On an Algorithm for the Solution of Generalized Prandtl Equations , 2001, Numerical Algorithms.

[4]  Gennadi Vainikko,et al.  Hypersingular Integral Equations and Their Applications , 2003 .

[5]  M. C. D. Bonis,et al.  On the simultaneous approximation of a Hilbert transform and its derivatives on the real semiaxis , 2017 .

[6]  Giuseppe Mastroianni,et al.  LAGRANGE INTERPOLATION IN SOME WEIGHTED UNIFORM SPACES , 2012 .

[7]  H. R. Kutt.  On the numerical evaluation of finite-part integrals involving an algebraic singularity , 1975 .

[8]  M. R. Capobianco,et al.  Uniform convergence of the collocation method for Prandtl's Integro-differential equation , 2000, The ANZIAM Journal.

[9]  F. Smithies,et al.  Singular Integral Equations , 1955, The Mathematical Gazette.

[10]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[11]  A. Timan.  Theory of Approximation of Functions of a Real Variable , 1994 .

[12]  E. Kanetsyan,et al.  On a method for solving Prandtl's integro‐differential equation applied to problems of continuum mechanics using polynomial approximations , 2017 .

[13]  G. Milovanović,et al.  Interpolation Processes: Basic Theory and Applications , 2008 .

[14]  U. Luther,et al.  Boundedness of the Hilbert transformation in some weighted Besov type spaces , 2000 .

[15]  R. Kanwal.  Linear Integral Equations , 1925, Nature.

[16]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[17]  Luisa Fermo.  Embedding theorems for functions with inner singularities , 2009 .

[18]  A. I. Kalandiya.  Mathematical Methods of Two-Dimensional Elasticity. , 1975 .

[19]  L. Dragoş.  Integration of Prandtl's equation with the aid of quadrature formulae of Gauss type , 1994 .

[20]  V. Sil’vestrov,et al.  The Prandtl integrodifferential equation and the contact problem for a piecewise homogeneous plate , 2010 .

[21]  V. Totik,et al.  Moduli of smoothness , 1987 .

[22]  L. Prandtl,et al.  The Mechanics of Viscous Fluids , 1935 .

[23]  Paul Nevai,et al.  Mean convergence of Lagrange interpolation. III , 1984 .

[24]  Maria Carmela De Bonis,et al.  Projection Methods and Condition Numbers in Uniform Norm for Fredholm and Cauchy Singular Integral Equations , 2006, SIAM J. Numer. Anal..

[25]  G. Mastroianni,et al.  Numerical Methods for Cauchy Singular Integral Equations in Spaces of Weighted Continuous Functions , 2005 .

[26]  M. Golberg.  The convergence of several algorithms for solving integral equations with finite part integrals. II , 1987 .

[27]  M. R. Capobianco,et al.  A fast algorithm for Prandtl's integro-differential equation , 1997 .

[28]  J. Radok,et al.  Singular Integral Equations: Boundary problems of functions theory and their applications to mathematical physics , 1977 .

[29]  Peter Junghanns,et al.  Cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution , 1997 .

[30]  Giuseppe Mastroianni,et al.  Lagrange Interpolation in Weighted Besov Spaces , 1999 .

[31]  B. Silbermann,et al.  Numerical Analysis for Integral and Related Operator Equations , 1991 .

[32]  Giovanni Monegato,et al.  Quadrature rules for Prandtl's integral equation , 1986, Computing.

[33]  W. Hackbusch.  Singular Integral Equations , 1995 .

[34]  Woula Themistoclakis,et al.  A numerical method for the generalized airfoil equation based on the de la Vallée Poussin interpolation , 2005 .

[35]  L. Dragoş.  A Collocation Method for the Integration of Prandtl's Equation , 1994 .

[36]  Maria Carmela De Bonis,et al.  Remarks on two integral operators and numerical methods for CSIE , 2014, J. Comput. Appl. Math..