Document Type : Original Article
Authors
University of Mazandaran
Abstract
In this article, we propose a new strategy for solving problems associated with weakly singular partial integro-differential equations. Our approach uses Multiple knot B-splines to develop a powerful arithmetical solution. We analyze the functional matrices used in this technique and provide a detailed overview of its functionality. additionally, we demonstrate the convergence of the proposed advance and verify its effectiveness via several numerical simulation.
Keywords
[1] Mohammad, M., & Trounev, A. (2020). Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications. Advances in Difference Equations, 2020(1), 618.
[2] Iserles, A. (2004). On the numerical quadrature of highly‐oscillating integrals I: Fourier transforms. IMA Journal of Numerical Analysis, 24(3), 365-391.
[3] Iserles, A. (2005). On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators. IMA journal of numerical analysis, 25(1), 25-44.
[4] Hieneman, M., Dunlap, G., & Kincaid, D. (2005). Positive support strategies for students with behavioral disorders in general education settings. Psychology in the Schools, 42(8), 779-794.
[5] De Boor, C., & De Boor, C. (1978). A practical guide to splines (Vol. 27, p. 325). New York: springer.
[6] Dehestani, H., Ordokhani, Y., & Razzaghi, M. (2020). Numerical solution of variable-order time fractional weakly singular partial integro-differential equations with error estimation. Mathematical Modelling and Analysis, 25(4), 680-701.
[7] Ghosh, B., & Mohapatra, J. (2023). Analysis of a second-order numerical scheme for time-fractional partial integro-differential equations with a weakly singular kernel. Journal of Computational Science, 74, 102157.
[8] Keshavarz, E., & Ordokhani, Y. (2019). A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels. Mathematical Methods in the Applied Sciences, 42(13), 4427-4443.
[9] Lyche, T., Manni, C., & Speleers, H. (2017). B-splines and spline approximation. Lecture notes.
[10] Kunoth, A., Lyche, T., Sangalli, G., Serra-Capizzano, S., Lyche, T., Manni, C., & Speleers, H. (2018). Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. Splines and PDEs: From Approximation Theory to Numerical Linear Algebra: Cetraro, Italy 2017, 1-76.
[11] Mouley, J., & Mandal, B. N. (2021). Wavelet‐based collocation technique for fractional integro‐differential equation with weakly singular kernel. Computational and Mathematical Methods, 3(4), e1158.
[12] Sadri, K., Hosseini, K., Baleanu, D., Ahmadian, A., & Salahshour, S. (2021). Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel. Advances in Difference Equations, 2021, 1-26.
[13] Schumaker, L. (2007). Spline functions: basic theory. Cambridge university press.
[14] Taghipour, M., & Aminikhah, H. (2022). A difference scheme based on cubic B-spline quasi-interpolation for the solution of a fourth-order time-fractional partial integro-differential equation with a weakly singular kernel. Sādhanā, 47(4), 253.
[15] Zhuang, P., Liu, F., Anh, V., & Turner, I. (2009). Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis, 47(3), 1760-1781.
)