Abstract
We present in this paper the convergence properties of Jacobi spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation. The solution is sufficiently smooth while the source function and the kernel function are smooth. We choose the Jacobi–Gauss points associated with the multidimensional Jacobi weight function \(\omega ({\mathbf{x}})=\Pi _{i=1}^d(1-x_i)^\alpha (1+x_i)^\beta ,\; -1<\alpha , \beta <\frac{1}{d}-\frac{1}{2}\) (d denotes the space dimensions) as the collocation points. The error analysis in \(L^\infty\)-norm and \(L_\omega ^2\)-norm theoretically justifies the exponential convergence of spectral collocation method in multidimensional space. We give two numerical examples in order to illustrate the validity of the proposed Jacobi spectral collocation method.
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Background
We observe that there are many numerical approaches for solving one-dimensional Volterra integral equation, such as Runge–Kutta method (Brunner 1984; Yuan and Tang 1990), polynomial collocation method (Brunner 1986; Brunner et al. 2001; Brunner and Tang 1989), multistep method (Mckee 1979; Houwen and Riele 1985), hp-discontinuous Galerkin method (Brunner and Schötzau 2006) and Taylor series method (Goldfine 1977). The spectral collocation method is the most popular form of the spectral methods among practitioners. It is convenient to implement for one-dimensional problems and generally leads to satisfactory results an long as the problems possess sufficient smoothness. In the literature (Tang et al. 2008), the authors proposed a Legendre spectral collocation method for Volterra integral equation with a regular kernel in one-dimensional space. Subsequently, Chen and Tang (2009, 2010), Chen et al. (2013), developed the spectral collocation method for one-dimensional weakly singular Volterra integral equation. The proofs of the convergence properties of spectral collocation method for Volterra integro-differential equation with a single spatial variable are given in Wei and Chen (2012a, b, 2013, 2014). Nevertheless, to the best of our knowledge, there have been no works regarding the theoretical analysis of the spectral approximation for multidimensional Volterra integral equation (Atdev and Ashirov 1977; Beesack 1985; Pachpatte 2011; Suryanarayana 1972), even for the case with smooth kernel.
We shall extend to several space dimensions the approximation results in Tang et al. (2008) for a single spatial variable. The expansion of Jacobi will be considered. We will be concerned with Sobolev-type norms that are most frequently applied to the convergence analysis of spectral methods. We get the discrete scheme by using multidimensional Gauss quadrature formula for the integral term. We will provide a rigorous verification of the exponential decay of the errors for approximate solution.
We study the multidimensional nonlinear Volterra integral equation of the form
by the Jacobi spectral collocation method. Here, \(g: [0,T_1]\times [0,T_2]\times \cdots \times [0,T_d] \rightarrow R\) and \(K: D\times R\rightarrow R\) (where \(D:=\{(t_1,s_1,t_2,s_2,\ldots ,t_d,s_d): 0\le s_i\le t_i\le T_i, i=1,2,\ldots ,d\)}) are given smooth functions. If the given functions are smooth on their respective domains, the solution y is also the smooth function (see Brunner 2004). This fact will be the standing point of this paper.
Discretization scheme
We consider now the domain \(\Omega =(-1,1)^d\) and we denote an element of \({\mathbb {R}}^d\) by \({\mathbf{x}}=(x_1,x_2,\ldots ,x_d)\). Let \(-1<\alpha , \beta <\frac{1}{d}-\frac{1}{2}\), if \(\omega =\omega ({\mathbf{x}})=\Pi _{i=1}^d(1-x_i)^\alpha (1+x_i)^\beta\) denotes a d-dimensional Jacobi weight function on \(\Omega\), we denote by \(L_\omega ^2(\Omega )\) the space of the measurable functions \(u:\Omega \rightarrow {\mathbb {R}}\) such that \(\int _\Omega |u({\mathbf{x}})|^2\omega ({\mathbf{x}})d{\mathbf{x}}<+\infty\). It is a Banach space for the norm
The space \(L_\omega ^2(\Omega )\) is a Hilbert space for the inner product
\(L^\infty (\Omega )\) is the Banach space of the measurable functions \(u:\Omega \rightarrow {\mathbb {R}}\) that are bounded outside a set of measure zero, equipped with the norm
Given a multi-index \(\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _d)\) of nonnegative integers, we set
and
We define \(H_\omega ^m(\Omega )\)= {\(v\in L_\omega ^2(\Omega )\): for each nonnegative multi-index \(\alpha\) with \(|\alpha |\le m\), the distributional derivative \(D^\alpha v\) belongs to \(L_\omega ^2(\Omega )\}.\) This is a Hilbert space for the inner product
which induces the norm
Let \(\{\tilde{x}_j, 0\le j\le N\}\) denote the Jacobi Gauss points on the one-dimensional interval \((-1,1)\) (see Canuto et al. 2006; Shen and Tang 2006). We now consider multidimensional Jacobi interpolation. Let \({\mathbb {P}}_N(\Omega )\) be the space of all algebraic polynomials of degree up to N in each variable \(x_i\) for \(i=1,2,\ldots ,d\). Let us introduce the Jacobi Gauss points in \(\Omega\):
and denote by \(I_N\) the interpolation operator at these points, i.e., for each continuous function u, \(I_Nu\in {\mathbb {P}}_N\) satisfies
We can represent \(I_Nu\) as follows:
where \(F_{\mathbf{j}}({\mathbf{x}})=F_{j_1}(x_1)F_{j_2}(x_2)\ldots F_{j_d}(x_d)\), \(\{F_j\}_{j=0}^N\) is the Lagrange interpolation basis function associated with the Jacobi collocation points \(\{\tilde{x}_j\}_{j=0}^N\). The multidimensional Jacobi Gauss quadrature formula is
We use the variable transformations \(t_i=\frac{T_i}{2}(1+x_i),\; x_i\in [-1,1]\) and \(s_i=\frac{T_i}{2}(1+\tau _i),\; \tau _i\in [-1,x_i],\;i=1,2,\ldots ,d\) to rewrite (1) as follows
Here,
and \(u(x_1,x_2,\ldots ,x_d)=y\left( \frac{T_1}{2}(1+x_1),\frac{T_2}{2}(1+x_2), \ldots ,\frac{T_d}{2}(1+x_d)\right)\) is the smooth solution of problem (3).
Firstly, Eq. (3) holds at the collocation points \(\tilde{{\mathbf{x}}}_{\mathbf{j}}=({\tilde{x}}_{j_1},{\tilde{x}}_{j_2}, \ldots ,{\tilde{x}}_{j_d})\) on \(\Omega\), i.e.,
In order to obtain high order accuracy for the problem (4), we transfer the integral domain \([-1,{\tilde{x}}_{j_1}]\times [-1,{\tilde{x}}_{j_2}]\cdots \times [-1,{\tilde{x}}_{j_d}]\) to a fixed interval \(\bar{\Omega }\)
by using the following transformation
where
Next, let \(u_{j_1j_2\cdots j_d}\) be the approximation of the function value \(u(\tilde{{\mathbf{x}}}_{\mathbf{j}})\) and use Legendre Gauss quadrature formula, (5) becomes
Here, \(\{\theta_{\mathbf{k}}, \Vert {\mathbf{k}}\Vert \le N\}\) denotes the Legendre Gauss points on the multidimensional space \(\Omega\) and \(\{{\omega }_{\mathbf{k}}, \Vert {\mathbf{k}}\Vert \le N\}\) denotes the corresponding weights. Let \(u_N(x_1,x_2,\ldots ,x_d)=\sum\nolimits _{\Vert {\mathbf{i}}\Vert \le N} u_{i_1i_2\ldots i_d}F_{i_1}(x_1)F_{i_2}(x_2)\ldots F_{i_d}(x_d)\). Now, we use \(u_N\) to approximate the solution u. Then, the Jacobi spectral collocation method is to seek \(u_N\) such that \(u_{i_1i_2\cdots i_d}\) satisfy the following collocation equation:
We can get the values of \(u_{i_1i_2\cdots i_d}\) by solving (8) and obtain the expressions of \(u_N({\mathbf{x}})\) accordingly.
Let the error function of the solution be written as \(e_u({\mathbf{x}}):=u({\mathbf{x}})-u_N({\mathbf{x}})\). Since the exact solution of the problem (1) can be written as \(y({\mathbf{t}})=u({\mathbf{x}})\; (t_i=\frac{T_i}{2}(1+x_i),\;t_i\in [0,T_i],\;x_i\in [-1,1])\), we can define its approximate solution \(y_N({\mathbf{t}})=u_N({\mathbf{x}})\). Then the corresponding error function satisfy
Remark
In our work, we let the multidimensional Jacobi weight function \(\omega ({\mathbf{x}})=\Pi _{i=1}^d(1-x_i)^\alpha (1+x_i)^\beta ,\; -1<\alpha , \beta <\frac{1}{d}-\frac{1}{2}\). So \(\omega (x)=(1-x)^\alpha (1+x)^\beta ,\; -1<\alpha , \beta <\frac{1}{2}\) for \(d=1\). In Tang et al. (2008), the authors choose \(\alpha =\beta =0\).
Some lemmas
The following result can be found in Canuto et al. (2006).
Lemma 1
Assume that Gauss quadrature formula is used to integrate the product \(u\phi\), where \(u\in H^m(\Omega )\) for some \(m> \frac{d}{2}\) and \(\phi \in {\mathbb {P}}_N(\Omega )\). Then there exists a constant C independent of N such that
where \((\cdot ,\cdot )\) represents the continuous inner product in \(L^2(\Omega )\) space and
The seminorm is defined as
Note that only pure derivatives in each spatial direction appear in this expression.
From Fedotov (2004), we have the following result on the Lebesgue constant for the Lagrange interpolation polynomials associated with the Jacobi-Gauss points.
Lemma 2
Let \(\Vert I_N\Vert _{\infty } :=\max \nolimits _{{\mathbf{x}}\in \bar{\Omega }}\sum \nolimits _{\Vert {\mathbf{k}}\Vert \le N}|F_{k_1}(x_1)F_{k_2}(x_2)\cdots F_{k_d}(x_d)|\), we have
Lemma 3
Assume that \(u({\mathbf{x}})\in H_\omega ^m(\Omega )\) for \(m>\frac{d}{2}\) and denote \((I_{N}u)({\mathbf{x}})\) its interpolation polynomial associated with the multidimensional Jacobi Gauss points \(\{\tilde{{\mathbf{x}}}_{\mathbf{j}},\Vert {\mathbf{j}}\Vert \le N\}\). Then the following estimates hold
Proof
The inequality (11) can be found in Canuto et al. (2006). We now prove (12). From Canuto et al. (2006), we have
We know that \(H_\omega ^l(\Omega )\) is embedded in \(C(\bar{\Omega })\) for \(l>\frac{d}{2}\), namely,
\(\square\)
The following Gronwall Lemma, whose proof can be found in Headley (1974), will be essential for establishing our main results.
Lemma 4
Suppose \(M\ge 0,\) a nonnegative integrable function \(E({\mathbf{x}})\) satisfies
where \(G({\mathbf{x}})\) is also an integrable function, we have
From Theorem 1 in Nevai (1984), we have the following mean convergence result of Lagrange interpolation based at the multidimensional Jacobi-Gauss points.
Lemma 5
For every bounded function \(v({\mathbf{x}})\), there exists a constant C independent of v such that
For \(r\ge 0\) and \(\kappa \in (0,1)\), \({\mathcal {C}}^{r,\kappa }(\bar{\Omega })\) will denote the space of functions whose r-th derivatives are \(H{\ddot{o}}lder\) continuous with exponent \(\kappa\), endowed with the norm:
We shall make use of a result of Ragozin (1970, (1971) in the following lemma.
Lemma 6
For nonnegative integer r and \(\kappa \in (0,1)\), there exists a constant \(C_{r,\kappa }>0\) such that for any function \(v\in {\mathcal {C}}^{r,\kappa }(\bar{\Omega })\), there exists a polynomial function \({\mathcal {T}}_Nv\in {\mathbb {P}}_N\) such that
Actually, \({\mathcal {T}}_N\) is a linear operator from \({\mathcal {C}}^{r,\kappa }(\bar{\Omega })\) into \({\mathbb {P}}_N\).
Lemma 7
Assume there are constants \(L_0, L_1,L_2,\ldots ,L_d\) such that
Let \(M_{v_1,v_2}\) be defined by
Then, for any \(\kappa \in (0,1)\) and \(v_1,v_2\in {\mathcal {C}}(\bar{\Omega })\), there exists a positive constant \(C\thicksim L_0, L_1,L_2,\ldots ,L_d\) such that
for any \({\mathbf{x}}^\prime , {\mathbf{x}}^{\prime \prime }\in \bar{\Omega }\) and \({\mathbf{x}}^\prime \ne {\mathbf{x}}^{\prime \prime }\). This implies that
Proof
For ease of exposition, and without essential loss of generality, we will proof this lemma for \(d=2\) and assume \(x_1^{\prime \prime }<x_1^\prime\), \(x_2^{\prime \prime }<x_2^\prime\),
Here,
where
similarly,
The estimate (18) for \(d = 2\) is obtained by combining (20)–(24). \(\square\)
Error estimates
Theorem 1
Let \(u({\mathbf{x}})\) be the exact solution of the multidimensional nonlinear Volterra integral equation (3), which is smooth. \(u_N({\mathbf{x}})\) is the approximate solution, i.e., \(u({\mathbf{x}})\approx u_N({\mathbf{x}}).\) Assume that
Then there is a constant C such that the errors satisfy for \(m>d+2\),
where
Proof
We subtract (8) from (5) to get the error equation
where
Using the variable transformation (6), we have
Multiplying \(F_{j_1}(x_1)F_{j_2}(x_2)\ldots F_{j_d}(x_d)\) on both sides of Eq. (27) and summing up \(\Vert {\mathbf{j}}\Vert \le N\) yield
Consequently,
where
It follows from the Gronwall inequality in Lemma 4 that
A straightforward computation shows that
Due to Lemma 3,
We now obtain the estimate for \(\Vert e_{u}\Vert _{L^\infty (\Omega )}\) by using (30)–(34),
where in last step we have used the following assumption,
This completes the proof of the theorem. \(\square\)
Theorem 2
If the hypotheses given in Theorem 1 hold and \(\kappa\) satisfies (35), then
Proof
By using (28) and Gronwall inequality in Lemma 4, we obtain that
Using Lemmas 1, 5 and (32) we have for
Due to Lemma 3,
The desired estimate (36) is obtained by combining (37)–(40) and using the same technique as in the proof of Theorem 1. \(\square\)
Numerical results
We give two numerical examples to confirm our analysis. To examine the accuracy of the results, \(L_\omega ^2\) and \(L^\infty\) errors are employed to assess the efficiency of the method. All the calculations are supported by the software Matlab.
Example 1
We consider the following two-dimensional Volterra integral equation
The corresponding exact solution is given by \(u(x,y)=e^{-\frac{xy}{2}}\). We select \(\alpha =-\frac{2}{3},\;\beta =-\frac{1}{2}\). Table 1 shows the errors \(\Vert u-u_N\Vert _{L_\omega ^2(\Omega )}\) and \(\Vert u-u_N\Vert _{L^\infty (\Omega )}\) obtained by using the spectral collocation method described above. Furthermore, the numerical results are plotted for \(2\le N\le 12\) in Fig. 1. It is observed that the desired exponential rate of convergence is obtained.
Example 2
Consider the equation with
The corresponding exact solution is given by \(v(x,y)=sin(x+y)\). We select \(\alpha =-\frac{2}{3},\;\beta =-\frac{3}{4}\). Table 2 shows the errors \(\Vert v-v_N\Vert _{L_\omega ^2(\Omega )}\) and \(\Vert v-v_N\Vert _{L^\infty (\Omega )}\). The numerical results are plotted for \(2\le N\le 12\) in Fig. 2.
Conclusions
In this paper, we proposed a spectral collocation method based on Jacobi orthogonal polynomials to obtain approximate solution for multidimensional nonlinear Volterra integral equation. The most important contribution of this work is that we are able to demonstrate rigorously that the errors of spectral approximations decay exponentially in both \(L^\infty (\Omega )\) norm and \(L^2_{\omega }(\Omega )\) norm on d-dimensional space, which is a desired feature for a spectral method.
References
Atdev S, Ashirov S (1977) Solutions of multidimensional nonlinear Volterra operator equations. Ukr Math J 29:437–442
Beesack PR (1985) Systems of multidimensional Volterra integral equations and inequalities. Nonlinear Anal Theory 9:1451–1486
Brunner H (1984) Implicit Runge–Kutta methods of optimal order for Volterra integro-differential equations. Math Comput 42:95–109
Brunner H (1986) Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J Numer Anal 6:221–239
Brunner H (2004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge
Brunner H, Tang T (1989) Polynomial spline collocation methods for the nonlinear Basset equation. Comput Math Appl 18:449–457
Brunner H, Schötzau D (2006) hp-discontinuous Galerkin time-stepping for Volterra integrodifferential equations. SIAM J Numer Anal 44:224–245
Brunner H, Pedas A, Vainikko G (2001) Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J Numer Anal 39:957–982
Canuto C, Hussaini MY, Quarteroni A et al (2006) Spectral methods fundamentals in single domains. Springer, Berlin
Chen Y, Tang T (2009) Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math 233:938–950
Chen Y, Tang T (2010) Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comput 79:147–167
Chen Y, Li X, Tang T (2013) A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions. J Comput Math 31:47–56
Fedotov AI (2004) Lebesgue constant estimation in multidimensional Sobolev space. J Math 14:25–32
Goldfine A (1977) Taylor series methods for the solution of Volterra integral and integro-differential equations. Math Comput 31:691–707
Headley VB (1974) A multidimensional nonlinear Gronwall inequality. J Math Anal Appl 47:250–255
Mckee S (1979) Cyclic multistep methods for solving Volterra integro-differential equations. SIAM J Numer Anal 16:106–114
Nevai P (1984) Mean convergence of Lagrange interpolation. Trans Am Math Soc 282:669–698
Pachpatte BG (2011) Multidimensional integral equations and inequalities. Springer, Berlin 9
Ragozin DL (1970) Polynomial approximation on compact manifolds and homogeneous spaces. Trans Am Math Soc 150:41–53
Ragozin DL (1971) Constructive polynomial approximation on spheres and projective spaces. Trans Am Math Soc 162:157–170
Shen J, Tang T (2006) Spectral and high-order methods with applications. Science Press, Beijing
Suryanarayana MB (1972) On multidimensional integral equations of Volterra type. Pac J Math 41:809–828
Tang T, Xu X, Chen J (2008) On spectral methods for Volterra integral equations and the convergence analysis. J Comput Math 26:825–837
van der Houwen PJ, te Riele HJJ (1985) Linear multistep methods for Volterra integral and integro-differential equations. Math Comput 45:439–461
Wei YX, Chen Y (2012a) Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Adv Appl Math Mech 4:1–20
Wei YX, Chen Y (2012b) Legendre spectral collocation methods for pantograph Volterra delay-integro-differential equation. J Sci Comput 53:672–688
Wei YX, Chen Y (2013) A spectral method for neutral Volterra integro-differential equation with weakly singular kernel. Numer Math Theory Methods Appl 6:424–446
Wei YX, Chen Y (2014) Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation. Appl Numer Math 81:15–29
Yuan W, Tang T (1990) The numerical analysis of implicit Runge–Kutta methods for a certain nonlinear integro-differential equation. Math Comput 54:155–168
Authors' contributions
YW and YC carried out the spectral collocation method studies, performed the error analysis and drafted the manuscript. XS participated in the numerical experiments. YZ helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by National Natural Science Foundation of China (11401347, 91430104, 11271145, 61401255, 11426193).
Competing interests
The authors declare that they have no competing interests.
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Wei, Y., Chen, Y., Shi, X. et al. Jacobi spectral collocation method for the approximate solution of multidimensional nonlinear Volterra integral equation. SpringerPlus 5, 1710 (2016). https://doi.org/10.1186/s40064-016-3358-z
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DOI: https://doi.org/10.1186/s40064-016-3358-z
Keywords
- Multidimensional nonlinear Volterra integral equation
- Jacobi collocation discretization
- Multidimensional Gauss quadrature formula
- Error estimates