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A combination method for solving multi-dimensional systems of Volterra integral equations with weakly singular kernels

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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.

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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

$$ u(\boldsymbol{x})\leq c+Bu(\boldsymbol{x}), $$

thus,

$$ \begin{array}{@{}rcl@{}} u(\boldsymbol{x})\leq c+Bu(\boldsymbol{x})\leq c+B(c+Bu(\boldsymbol{x}))\leq...\leq {\sum}_{k=0}^{n-1}B^{k}c+B^{n}u(\boldsymbol{x}). \end{array} $$

When n = 2, using Fubini’s theorem and inequality \((a-t)(t-b)\leq (\frac {a-b}{2})^{2},\ t\in [b,a]\), then

$$ \begin{array}{@{}rcl@{}} B^{2}u(\boldsymbol{x})&=&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{0}^{t_{1}}\cdots{\int}_{0}^{t_{s}}{\prod}_{m=1}^{s}(x_{m}-t_{m})^{-\alpha_{m}}(t_{m}-\tau_{m})^{-\alpha_{m}}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}\mathrm{d}\boldsymbol{t}\\ &=&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{\tau_{1}}^{x_{1}}\cdots{\int}_{\tau_{s}}^{x_{s}}{\prod}_{m=1}^{s}(x_{m}-t_{m})^{-\alpha_{m}}(t_{m}-\tau_{m})^{-\alpha_{m}}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{\tau}\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{\tau_{1}}^{x_{1}}\cdots{\int}_{\tau_{s}}^{x_{s}}{\prod}_{m=1}^{s}\frac{\left( x_{m}-\tau_{m}\right)^{-2\alpha_{m}}}{2^{-2\alpha_{m}}}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{\tau}\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\prod}_{m=1}^{s}\frac{\left( x_{m}-\tau_{m}\right)^{2(1-\alpha_{m})-1}}{2^{-2\alpha_{m}}}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}. \end{array} $$

When n = 3,

$$ \begin{array}{@{}rcl@{}} B^{3}u(\boldsymbol{x})&=&B(B^{2}u(\boldsymbol{x}))\leq B\left( {\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{2(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}\right)\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{0}^{t_{1}}\cdots{\int}_{0}^{t_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-t_{m})^{-\alpha_{m}}(t_{m}-\tau_{m})^{2(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}\mathrm{d}\boldsymbol{t}\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{\tau_{1}}^{x_{1}}\cdots{\int}_{\tau_{s}}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-t_{m})^{-\alpha_{m}}(t_{m}-\tau_{m})^{2(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{\tau}\\ &\leq& {\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{\tau_{1}}^{x_{1}}\cdots{\int}_{\tau_{s}}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{-2\alpha_{m}}(t_{m}-\tau_{m})^{1-\alpha_{m}}}{(2^{-2\alpha_{m}})^{2}}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{\tau}\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{3(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{2}(2-\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}. \end{array} $$

When n = 4,

$$ \begin{array}{@{}rcl@{}} B^{4}u(\boldsymbol{x})&=&B(B^{3}u(\boldsymbol{x}))\leq B\left( {\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{3(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{2}(2-\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}\right)\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{0}^{t_{1}}\cdots{\int}_{0}^{t_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-t_{m})^{-\alpha_{m}}(t_{m}-\tau_{m})^{3(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{2}(2-\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}\mathrm{d}\boldsymbol{t}\\ &\leq& {\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{\tau_{1}}^{x_{1}}\cdots{\int}_{\tau_{s}}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{-2\alpha_{m}}(t_{m}-\tau_{m})^{2(1-\alpha_{m})}}{(2^{-2\alpha_{m}})^{3}(2-\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{\tau}\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{4(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{3}(2-\alpha_{m})(3-2\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}. \end{array} $$

When n ≥ 4, suppose that Bnu(x) has the following estimate

$$ \begin{array}{@{}rcl@{}} B^{n}u(\boldsymbol{x}) &\leq{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{n(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{n-1}\prod\limits_{i=2}^{n-1}(i-(i-1)\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}. \end{array} $$

Then, we can obtain the following inequality

$$ \begin{array}{@{}rcl@{}} &&B^{n+1}u(\boldsymbol{x})=B(B^{n}u(\boldsymbol{x}))\\ &\leq& B\left( {\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}\prod\limits_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{n(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{n-1}\prod\limits_{i=2}^{n-1}(i-(i-1)\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}\right)\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{0}^{t_{1}}\cdots{\int}_{0}^{t_{s}}\prod\limits_{m=1}^{s}\frac{(x_{m}-t_{m})^{-\alpha_{m}}(t_{m}-\tau_{m})^{n(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{n-1}{\prod}_{i=2}^{n-1}(i-(i-1)\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}\mathrm{d}\boldsymbol{t}\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{\tau_{1}}^{x_{1}}\cdots{\int}_{\tau_{s}}^{x_{s}}\prod\limits_{m=1}^{s}\frac{(x_{m}-t_{m})^{-\alpha_{m}}(t_{m}-\tau_{m})^{n(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{n-1}\prod\limits_{i=2}^{n-1}(i-(i-1)\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{\tau}\\ &\leq& {\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\int}_{\tau_{1}}^{x_{1}}\cdots{\int}_{\tau_{s}}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{-2\alpha_{m}}(t_{m}-\tau_{m})^{(n-1)(1-\alpha_{m})}}{(2^{-2\alpha_{m}})^{n}\prod\limits_{i=2}^{n-1}(i-(i-1)\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{t}\mathrm{d}\boldsymbol{\tau}\\ &\leq&{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}\prod\limits_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{(n+1)(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{n}\prod\limits_{i=2}^{n}(i-(i-1)\alpha_{m})}u(\boldsymbol{\tau})\mathrm{d}\boldsymbol{\tau}. \end{array} $$

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

$$ \begin{array}{@{}rcl@{}} B^{n}c &\leq{\int}_{0}^{x_{1}}\cdots{\int}_{0}^{x_{s}}{\prod}_{m=1}^{s}\frac{(x_{m}-\tau_{m})^{n(1-\alpha_{m})-1}}{(2^{-2\alpha_{m}})^{n-1}\prod\limits_{i=2}^{n-1}(i-(i-1)\alpha_{m})}c\mathrm{d}\boldsymbol{\tau}\\ &\leq \prod\limits_{m=1}^{s}\frac{c}{n(1-\alpha_{m})(2^{-2\alpha_{m}})^{n-1}{\prod}_{i=2}^{n-1}(i-(i-1)\alpha_{m})}, \end{array} $$

thus,

$$ \begin{array}{@{}rcl@{}} u(\boldsymbol{x})&\leq {\sum}_{k=0}^{n-1}B^{k}c+B^{n}u(\boldsymbol{x})\\ &\leq c+{\sum}_{k=1}^{\infty}\prod\limits_{m=1}^{s}\frac{c}{k(1-\alpha_{m})(2^{-2\alpha_{m}})^{k-1}\prod\limits_{i=2}^{k-1}(i-(i-1)\alpha_{m})}. \end{array} $$

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

$$ \begin{array}{@{}rcl@{}} &\left|\left|K_{1}Y(\boldsymbol{\sigma})\right|\right|_{\infty}\\ &=\left|\left|\left[\begin{array}{cccc} a_{1}(\boldsymbol{\sigma})+\sum\limits_{j=1}^{M}{\int}_{0}^{\sigma_{1}}\cdots{\int}_{0}^{\sigma_{s}}\prod\limits_{m=1}^{s}(\sigma_{m}-\tau_{m})^{-\alpha_{1_{jm}}}h_{1j}(\boldsymbol{\sigma};\boldsymbol{\tau};y_{j}(\boldsymbol{\tau}))\mathrm{d}\boldsymbol{\tau} \\ a_{2}(\boldsymbol{\sigma})+\sum\limits_{j=1}^{M}{\int}_{0}^{\sigma_{1}}\cdots{\int}_{0}^{\sigma_{s}}\prod\limits_{m=1}^{s}(\sigma_{m}-\tau_{m})^{-\alpha_{2_{jm}}}h_{2j}(\boldsymbol{\sigma};\boldsymbol{\tau};y_{j}(\boldsymbol{\tau}))\mathrm{d}\boldsymbol{\tau} \\ {\vdots} \\ a_{M}(\boldsymbol{\sigma})+\sum\limits_{j=1}^{M}{\int}_{0}^{\sigma_{1}}\cdots{\int}_{0}^{\sigma_{s}}\prod\limits_{m=1}^{s}(\sigma_{m}-\tau_{m})^{-\alpha_{M_{jm}}}h_{Mj}(\boldsymbol{\sigma};\boldsymbol{\tau};y_{j}(\boldsymbol{\tau}))\mathrm{d}\boldsymbol{\tau} \end{array}\right]\right|\right|_{\infty}\\ &=\max\limits_{1\leq q\leq M} \left\{\left|\left|a_{q}(\boldsymbol{\sigma})+\sum\limits_{j=1}^{M}{\int}_{0}^{\sigma_{1}}\cdots{\int}_{0}^{\sigma_{s}}\prod\limits_{m=1}^{s}(\sigma_{m}-\tau_{m})^{-\alpha_{q_{jm}}}h_{qj}(\boldsymbol{\sigma};\boldsymbol{\tau};y_{j}(\boldsymbol{\tau}))\mathrm{d}\boldsymbol{\tau}\right|\right|_{\infty}\right\}\\ &\leq \widetilde{A}+ \overline{H}M\left|\left|{\int}_{0}^{\sigma_{1}}\cdots{\int}_{0}^{\sigma_{s}}\prod\limits_{m=1}^{s}(\sigma_{m}-\tau_{m})^{-\alpha_{q_{jm}}}\mathrm{d}\boldsymbol{\tau}\right|\right|_{\infty}\\ &\leq \widetilde{A}+\frac{\overline{H}M}{\overline{\alpha}}<\infty, \end{array} $$

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 σiωi, i = 1,...,s, we have

$$ \begin{array}{@{}rcl@{}} &\left|\left|K_{1}Y(\boldsymbol{\sigma})-K_{1}Y(\boldsymbol{\omega})\right|\right|_{\infty}\\ &=\max\limits_{1\leq q\leq M} \bigg\{\bigg|\bigg|a_{q}(\boldsymbol{\sigma})+\sum\limits_{j=1}^{M}{\int}_{0}^{\sigma_{1}}\cdots{\int}_{0}^{\sigma_{s}}\prod\limits_{m=1}^{s}(\sigma_{m}-\tau_{m})^{-\alpha_{q_{jm}}}h_{qj}(\boldsymbol{\sigma};\boldsymbol{\tau};y_{j}(\boldsymbol{\tau}))\mathrm{d}\boldsymbol{\tau} \\ &\quad-a_{q}(\boldsymbol{\omega})-\sum\limits_{j=1}^{M}{\int}_{0}^{\omega_{1}}\cdots{\int}_{0}^{\omega_{s}}\prod\limits_{m=1}^{s}(\omega_{m}-\upsilon_{m})^{-\alpha_{q_{jm}}}h_{qj}(\boldsymbol{\omega};\boldsymbol{\upsilon};y_{j}(\boldsymbol{\upsilon}))\mathrm{d}\boldsymbol{\upsilon} \bigg|\bigg|_{\infty}\bigg\}. \end{array} $$

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.

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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

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