Abstract
In this paper, we proposed an efficient approach for solving the multi-dimensional systems of weakly singular Volterra integral equations (SVIEs). The solution of these equations may be smooth or non-smooth, because its derivatives may be bounded or unbounded at the left endpoint of the interval of integration. In order to avoid the low-order accuracy caused by the singularity of the solution at the boundary of the integration domain, some smooth transformations are used to convert the original equation into a new equation with a more smooth solution. Then, the transformed equation can be efficiently solved by using Euler polynomials combined with Gauss-Jacobi quadrature formula, and then, the numerical solution of the original equation can be obtained through some inverse transformations. In addition, the existence and uniqueness of the solution of the system of original equations and approximate equations are proved by Gronwall inequality and the collectively compact theory, respectively. We also give the convergence analysis and error estimate of the proposed method. Finally, some numerical examples are provided to illustrate the efficiency of the method.
Similar content being viewed by others
References
Atkinson, K., Flores, J.: The discrete collocation method for nonlinear integral equations. IMA J. Numer. Anal. 13, 195–213 (1993)
Anselmi-Tamburini, U., Spinolo, G.: On the least-squares determinations of lattice dimensions: A modified singular value decomposition approach to ill-conditioned cases. J. Appl. Cryst. 26, 5–8 (1993)
Assari, P., Dehghan, M.: The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision. Eng. Comput. 33, 853–870 (2017)
Assari, P., Dehghan, M.: A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions. Eur. Phys. J. Plus., 132 (2017)
Ahmood, W.A., Kilicman, A.: Solutions of linear multi-dimensional fractional order Volterra integral equations. J. Theor. Appl. Inform. Technol. 89 (2), 381–388 (2016)
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations, p 15. Cambridge University Press, Cambridge (2004)
Baratella, P.: A nyström interpolant for some weakly singular nonlinear Volterra integral equations. J. Comput. Appl. Math. 237, 542–555 (2013)
Balcı, M.A., Sezer, M.: Hybrid Euler-Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations. Appl. Math. Comput. 273, 33–41 (2016)
Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM. J. Numer. Anal. 41, 364–381 (2003)
Chen, C., He, X.M., Huang, J.: Mechanical quadrature methods and their extrapolations for solving the first kind boundary integral equations of Stokes equation. Appl. Numer. Math. 96, 165–179 (2015)
Cai, H., Chen, Y.: A fractional order collocation method for second kind Volterra integral equations with weakly singular kernels. J. Sci. Comput. 75, 970–992 (2018)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, p 375. Springer, Berlin (1996)
Hansen, P.C., O’Leary, D.P.: The use of the l-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993)
Huang, J., Lv, T., Li, Z.C.: Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs. Appl. Numer. Math. 59(12), 2908–2922 (2009)
Huang, C., Tang, T., Zhang, Z.: Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions. J. Comput. Math. 29(6), 698–719 (2011)
Hashemizadeh, E., Ebadi, M.A., Noeiaghdam, S.: Matrix method by Genocchi polynomials for solving nonlinear Volterra integral equations with weakly singular kernels. Symmetry, 12 (2020)
Kant, K., Nelakanti, G.: Approximation methods for second kind weakly singular Volterra integral equations. J. Comput. Appl. Math. 368, 112531 (2020)
Lv, T., Huang, J.: High Precision Algorithm for Integral Equation. China Science Publishing (2013)
Li, H., Huang, J.: High-accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems. Comput. Math. Appl. 71(1), 459–469 (2016)
Liu, H.Y., Huang, J., Pan, Y.B., Zhang, J.P.: Barycentric interpolation collocation methods for solving linear and nonlinear high-dimensional Fredholm integral equations. J. Comput. Appl. Math. 327, 141–154 (2018)
Liu, H.Y., Huang, J., Zhang, W., Ma, Y.Y.: Meshfree approach for solving multi-dimensional systems of Fredholm integral equations via barycentric Lagrange interpolation. Appl. Math. Comput. 346, 295–304 (2019)
Liang, H., Brunner, H.: The fine error estimation of collocation methods on uniform meshes for weakly singular Volterra integral equations. J. Sci. Comput. 84, 12 (2020)
Mirzaee, F., Bimesl, S.: A new approach to numerical solution of second-order linear hyperbolic partial differential equations arising fom physics and engineering. Result Phys. 3, 241–247 (2013)
Mirzaee, F., Bimesl, S.: Application of euler matrix method for solving linear and a class of nonlinear fredholm integro-differential equations. Mediterr. J. Math 11, 999–1018 (2014)
Mirzaee, F., Bimesl, S.: A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients. J. Egypt. Math. Soci 22, 238–248 (2014)
Mirzaee, F., Bimesl, S.: Solving systems of high-order linear differential-difference equations via Euler matrix method. J. Egypt. Math. Soci 23, 286–291 (2015)
Mirzaee, F., Bimesl, S.: Numerical solutions of systems of high-order Fredholm integro-differential equations using Euler polynomials. Appl. Math. Model 39, 6767–6779 (2015)
Mirzaee, F., Bimesl, S.: A uniformly convergent euler matrix method for telegraph equations having constant coefficients. Mediterr. J. Math. 13, 497–515 (2016)
Mirzaee, F., Bimesl, S., Tohidi, E.: Solving nonlinear fractional integro-differential equations of Volterra type using novel mathematical matrices. J. Comput. Nonlin. Dyn 10, 061016 (2015)
Mirzaee, F, Samadyar, N, Hoseini, S.F: A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients. J. Comput. Appl. Math 330, 574–585 (2018)
Maleknejad, K., Ostadi, A.: Numerical solution of system of Volterra integral equations with weakly singular kernels and its convergence analysis. Appl. Numer. Math. 115, 82–98 (2017)
Micula, S.: A fast converging iterative method for Volterra integral equations of the second kind with delayed arguments. Fixed Point Theory 16(2), 371–380 (2015)
Micula, S.: On some iterative numerical methods for a Volterra functional integral equation of the second kind. J. Fix. Point. Theory. Appl. 19, 1815–1824 (2017). https://doi.org/10.1007/s11784-016-0336-6
Micula, S.: A numerical method for weakly singular nonlinear Volterra integral equations of the second kind. Symmetry Basel. 12, 1862 (2020). https://doi.org/10.3390/sym12111862
Pan, Y.B., Huang, J., Ma, Y.Y.: Bernstein series solutions of multidimensional linear and nonlinear Volterra integral equations with fractional order weakly singular kernels. Appl. Math. Comput. 347, 149–161 (2019)
Rivlin, T.J.: An Introduction to the Approximation of Functions. Dover Publications (1969)
Shen, J., Tang, T., Wang, L.L.: Spectral Methods. Springer Series in Computational Mathematics Heidelberg: Springer (2011)
Shi, X.L., Wei, Y.X., Huang, F.L.: Spectral collocation methods for nonlinear weakly singular Volterra integro-differential equations. Numer. Meth. Part. D. E. 35, 576–596 (2019)
Tikhonov, A., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, New York (1977)
Varah, J.M.: On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems. SIAM J. Numer. Anal. 10(2), 257–267 (1973)
Vainikko, G.: Multidimensional Weakly Singular Integral Equations. Springer, New York (2006)
Wang, Y.X., Zhu, L.: Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method. Adv. Differ. Equ., 27 (2017)
Wang, C.L., Wang, Z.Q., Jia, H.L.: An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels. J. Sci. Comput. 72, 647–678 (2017)
Wang, L., Tian, H.J., Yi, L.J.: An hp-version of the discontinuous Galerkin time-stepping method for Volterra integral equations with weakly singular kernels. Appl. Numer. Math. 161, 218–232 (2021)
Yang, Y., Kang, S.J., Vasil’ev, V.I.: The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electron. Res. Arch. 28(3), 1161–1189 (2020)
Zaky, M.A., Ameen, I.G.: A novel Jacobi spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions. Eng Comput-Germany (2020)
Zaky, M.A.: An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl. Numer. Math. 154, 205–222 (2020)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix: 1
In this appendix, we give a proof of Lemma 1. From inequality (3.1), we know that (3.2) holds when u(x) = 0. Next, we consider the case of u(x)≠ 0, let \(Bu(\boldsymbol {x})={\int \limits }_{0}^{x_{1}}\cdots {\int \limits }_{0}^{x_{s}}\prod \limits _{m=1}^{s}(x_{m}-t_{m})^{-\alpha _{m}}u(\boldsymbol {t})\mathrm {d}\boldsymbol {t}\), for nonnegative integrable function u(x). Then, we can write (3.1) as
thus,
When n = 2, using Fubini’s theorem and inequality \((a-t)(t-b)\leq (\frac {a-b}{2})^{2},\ t\in [b,a]\), then
When n = 3,
When n = 4,
When n ≥ 4, suppose that Bnu(x) has the following estimate
Then, we can obtain the following inequality
According to mathematical induction method, the estimate of Bnu(x) holds and is easy to verify as \(n\rightarrow \infty \), \(B^{n}u(\boldsymbol {x})\rightarrow 0\). When u(x) = c, we have
thus,
The proof of the Lemma 1 is complete.
Appendix: 2
Let \(\widetilde {A}=\max \limits \left \{||a_{q}(\boldsymbol {\sigma })||_{\infty }\right \}_{q=1}^{M}\) and \(\overline {H}=\max \limits \left \{||h_{qj}(\boldsymbol {\sigma };\boldsymbol {\tau };y_{j}(\boldsymbol {\tau }))||_{\infty }\right \}_{q,j=1}^{M}\). In order to prove K1 is a compact operator, we need to prove K1B is a relatively compact set, where \(B=\{Y(\boldsymbol {\sigma }):||Y||_{\infty }\leq 1\}\) is a unit ball. According to Ascoli-Arzela’s Theorem, we just need to prove that the function K1Y (σ) ∈ K1B is uniformly bounded and equi-continuous. For an arbitrary Y (σ) ∈ B, we have
where \(\overline {\alpha }=\min \limits _{1\leq q,j\leq M} \left \{\prod \limits _{m=1}^{s}(1-\alpha _{q_{jm}})\right \}\), so, K1Y (σ) ∈ K1B is uniformly bounded. For arbitrary σ,ω ∈ [0, 1]s and assume that σi ≥ ωi, i = 1,...,s, we have
Because \(\left |\left |K_{1}Y(\boldsymbol {\sigma })-K_{1}Y(\boldsymbol {\omega })\right |\right |_{\infty }\) is convergent to 0 when \(\left |\left |\boldsymbol {\sigma }-\boldsymbol {\omega }\right |\right |_{\infty }\rightarrow 0\), K1Y (σ) ∈ K1B is equi-continuous, that is, K1B is a relatively compact set. Therefore, K1 is a compact operator.
Rights and permissions
About this article
Cite this article
Wang, Y., Huang, J., Zhang, L. et al. A combination method for solving multi-dimensional systems of Volterra integral equations with weakly singular kernels. Numer Algor 91, 473–504 (2022). https://doi.org/10.1007/s11075-022-01270-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-022-01270-6
Keywords
- Euler polynomial
- Weakly singular integral equation
- Smoothing transformation
- Gauss-Jacobi quadrature formula
- Convergence analysis