Abstract
This paper presents a new trigonometric composite Hermite interpolation method for solving Fredholm linear integral equations. This operator approximates locally both the function and its derivative, which is known on the subdivision nodes. Then we derive a class of quadrature rules with endpoint corrections based on integrating the composite Hermite interpolant. We also provide error estimation and numerical examples to illustrate that this new operator can provide highly accurate results.
References
[1] M. Ajeddar and A. Lamnii, Smooth reverse subdivision of uniform algebraic hyperbolic B-splines and wavelets, Int. J. Wavelets Multiresolut. Inf. Process. 19 (2021), no. 5, Paper No. 2150018. 10.1142/S0219691321500181Search in Google Scholar
[2] C. Allouch and P. Sablonnière, Iteration methods for Fredholm integral equations of the second kind based on spline quasi-interpolants, Math. Comput. Simulation 99 (2014), 19–27. 10.1016/j.matcom.2013.04.014Search in Google Scholar
[3] C. Allouch, P. Sablonnière and D. Sbibih, Solving Fredholm integral equations by approximating kernels by spline quasi-interpolants, Numer. Algorithms 56 (2011), no. 3, 437–453. 10.1007/s11075-010-9396-7Search in Google Scholar
[4] C. Allouch, D. Sbibih and M. Tahrichi, Superconvergent Nyström and degenerate kernel methods for Hammerstein integral equations, J. Comput. Appl. Math. 258 (2014), 30–41. 10.1016/j.cam.2013.08.025Search in Google Scholar
[5] C. Allouch, D. Sbibih and M. Tahrichi, Superconvergent product integration method for Hammerstein integral equations, J. Integral Equations Appl. 31 (2019), no. 1, 1–28. 10.1216/JIE-2019-31-1-1Search in Google Scholar
[6] K. Atkinson and G. Chandler, The collocation method for solving the radiosity equation for unoccluded surfaces, J. Integral Equations Appl. 10 (1998), no. 3, 253–290. 10.1216/jiea/1181074231Search in Google Scholar
[7]
D. Barrera, S. Eddargani, A. Lamnii and M. Oraiche,
On nonpolynomial monotonicity-preserving
[8] D. Barrera, F. Elmokhtari and D. Sbibih, Two methods based on bivariate spline quasi-interpolants for solving Fredholm integral equations, Appl. Numer. Math. 127 (2018), 78–94. 10.1016/j.apnum.2017.12.016Search in Google Scholar
[9] A. Bellour, D. Sbibih and A. Zidna, Two cubic spline methods for solving Fredholm integral equations, Appl. Math. Comput. 276 (2016), 1–11. 10.1016/j.amc.2015.11.055Search in Google Scholar
[10] A. H. Borzabadi and O. S. Fard, A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind, J. Comput. Appl. Math. 232 (2009), no. 2, 449–454. 10.1016/j.cam.2009.06.038Search in Google Scholar
[11]
C. Conti and R. Morandi,
Piecewise
[12] H. B. Curry and I. J. Schoenberg, On spline distributions and their limits-the polya distribution functions, Amer. Math. Soc. 53 (1947), Paper No. 1114. Search in Google Scholar
[13] S. Eddargani, A. Lamnii and M. Lamnii, On algebraic trigonometric integro splines, ZAMM Z. Angew. Math. Mech. 100 (2020), no. 2, Article ID e201900262. 10.1002/zamm.201900262Search in Google Scholar
[14] S. Eddargani, A. Lamnii, M. Lamnii, D. Sbibih and A. Zidna, Algebraic hyperbolic spline quasi-interpolants and applications, J. Comput. Appl. Math. 347 (2019), 196–209. 10.1016/j.cam.2018.08.018Search in Google Scholar
[15] S. Kumar and I. H. Sloan, A new collocation-type method for Hammerstein integral equations, Math. Comp. 48 (1987), no. 178, 585–593. 10.1090/S0025-5718-1987-0878692-4Search in Google Scholar
[16] A. Lahtinen, Shape preserving interpolation by quadratic splines, J. Comput. Appl. Math. 29 (1990), no. 1, 15–24. 10.1016/0377-0427(90)90191-2Search in Google Scholar
[17] A. Lamnii, M. Lamnii and F. Oumellal, Computation of Hermite interpolation in terms of B-spline basis using polar forms, Math. Comput. Simulation 134 (2017), 17–27. 10.1016/j.matcom.2016.09.009Search in Google Scholar
[18]
C. Manni,
[19] G. Mastroianni, G. V. Milovanović and D. Occorsio, Nyström method for Fredholm integral equations of the second kind in two variables on a triangle, Appl. Math. Comput. 219 (2013), no. 14, 7653–7662. 10.1016/j.amc.2013.01.054Search in Google Scholar
[20] S. Micula and G. Micula, On the superconvergent spline collocation methods for the Fredholm integral equations on surfaces, Math. Balkanica (N. S.) 19 (2005), no. 1–2, 155–166. Search in Google Scholar
[21] M. S. Mummy, Hermite interpolation with B-splines, Comput. Aided Geom. Design 6 (1989), no. 2, 177–179. 10.1016/0167-8396(89)90021-6Search in Google Scholar
[22] P. Sablonnière, D. Sbibih and M. Tahrichi, High-order quadrature rules based on spline quasi-interpolants and application to integral equations, Appl. Numer. Math. 62 (2012), no. 5, 507–520. 10.1016/j.apnum.2011.12.004Search in Google Scholar
[23] I. J. Schoenberg and A. Sharma, Cardinal interpolation and spline functions. V. The B-splines for cardinal Hermite interpolation, Linear Algebra Appl. 7 (1973), 1–42. 10.1016/0024-3795(73)90034-7Search in Google Scholar
[24] L. L. Schumaker, On shape preserving quadratic spline interpolation, SIAM J. Numer. Anal. 20 (1983), no. 4, 854–864. 10.1137/0720057Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Around the Świątkowski-type conditions
- Metamorphism—an integral transform reducing the order of a differential equation
- Linear isomorphic spaces of Cesàro–Nörlund operator, their duals and matrix transformations
- A generalized Suzuki–Berinde contraction that characterizes Banach spaces
- Asymptotic behavior of solutions toward the constant state to the Cauchy problem for the non-viscous diffusive dispersive conservation law
- Trigonometric Hermite interpolation method for Fredholm linear integral equations
- Smoothing Levenberg–Marquardt algorithm for solving non-Lipschitz absolute value equations
- Two-dimensional EMD with shape-preserving spline interpolation
- Computing subdifferential limits of operators on Banach spaces
- Certain aspects of ℐ-statistical supremum and ℐ-statistical infimum of real-valued sequences
- Weighted Simpson-like type inequalities for quasi-convex functions
- Bochner formula in generalized (k,μ)-space forms
- Convergence of a conjugate function in Zygmund space by almost Nörlund transform
- 2-point left Radau-type inequalities via s-convexity
- A general size-biased distribution
- A study of fuzzy anti-λ-ideal convergent triple sequence spaces
- Euler-type integrals for the generalized hypergeometric matrix function
- Korn’s inequality in anisotropic Sobolev spaces
Articles in the same Issue
- Frontmatter
- Around the Świątkowski-type conditions
- Metamorphism—an integral transform reducing the order of a differential equation
- Linear isomorphic spaces of Cesàro–Nörlund operator, their duals and matrix transformations
- A generalized Suzuki–Berinde contraction that characterizes Banach spaces
- Asymptotic behavior of solutions toward the constant state to the Cauchy problem for the non-viscous diffusive dispersive conservation law
- Trigonometric Hermite interpolation method for Fredholm linear integral equations
- Smoothing Levenberg–Marquardt algorithm for solving non-Lipschitz absolute value equations
- Two-dimensional EMD with shape-preserving spline interpolation
- Computing subdifferential limits of operators on Banach spaces
- Certain aspects of ℐ-statistical supremum and ℐ-statistical infimum of real-valued sequences
- Weighted Simpson-like type inequalities for quasi-convex functions
- Bochner formula in generalized (k,μ)-space forms
- Convergence of a conjugate function in Zygmund space by almost Nörlund transform
- 2-point left Radau-type inequalities via s-convexity
- A general size-biased distribution
- A study of fuzzy anti-λ-ideal convergent triple sequence spaces
- Euler-type integrals for the generalized hypergeometric matrix function
- Korn’s inequality in anisotropic Sobolev spaces