Numerical Solution of a Non-Linear Volterra Integral Equation

Abstract

In this paper, a numerical method to solve non-linear integral equations based on a successive approximation technique is considered. A sequence of functions is produced which converges to the solution. The process includes a fixed point method, a quadrature rule, and an interpolation method. To find a total bound of the error, we investigate error bounds for each approximation and by combining them, we will derive an estimate for the total error. The accuracy and efficiency of the method is illustrated in some numerical examples.

Appendix: Stability and Convergence of Cubic Spline Interpolation

Theorem 11

For an interval partitioning

$$0=t_{0}<t_{1}<\cdots<t_{N}=1,\quad\tau_{i}:=[t_{i-1},t_{i}],\quad h_{\tau_{i}}:=t_{i}-t_{i-1} $$

with corresponding function values $\mathbf {f}:=(f_{i})_{i=0}^{N}$, let u _Δ denote the cubic interpolating spline with boundary conditions

$$u_{\Delta}^{\prime}(0)=\frac{f_{2}-f_{0}}{t_{2}-t_{0}}\quad\text{ and }\quad u_{\Delta}^{\prime}(1) =\frac{f_{N}-f_{N-2}}{t_{N}-t_{N-2}}. $$

• a. Then

  $$ \left\|u_{\Delta}\right\|_{C_{0}(I) }\leq C\|\mathbf{f}\|_{\max}, $$

  (55)

  where C depends on the global quasi-uniformity h/h _min with h _min := min{h _τ : τ ∈ Δ}. Let f ∈ C ^ℓ(I) for some 0 ≤ ℓ ≤ 4. Then

  $$ \|f-u_{\Delta}\| \leq Ch^{\ell}\left\|f^{(\ell)}\right\|. $$

  (56)

• b. Let the constant of local quasi-uniformity be given by

  $$ \kappa:=\max_{1\leq i\leq N-1}\left\{\max\left\{\frac{h_{\tau_{i}}}{h_{\tau_{i+1}}},\frac{h_{\tau_{i+1}}}{h_{\tau_{i}}}\right\}\right\} $$

  (57)

  and assume κ ² <2. Then

  $$ \left\| u_{\Delta}\right\|_{C^{0}(I)}\leq C_{\Delta}\|\mathbf{f}\|_{\max}\quad\text{ with }~C_{\Delta}\leq\frac{C}{2-\kappa^{2}}, $$

  (58)

  where C only depends on κ. Let f ∈ C ^ℓ(I) for some 0 ≤ ℓ ≤ 4. Then

  $$ \left\| f-u_{\Delta}\right\| \leq\frac{C}{2-\kappa^{2}}h^{\ell}\left\|f^{(\ell)}\right\|. $$

  (59)

Proof

For $k,p\in \mathbb {N}_{0}$, let

$$S^{k,p}:=\left\{v\in C^{k}(I)\mid \forall\tau_{i},~ v|_{\tau_{i}}\in\mathbb{P}_{p}\right\}\quad\text{ and }\quad S_{0}^{k,p}:=\left\{v\in S^{k,p}:v(0)=v(1)=0\right\}. $$

For simplicity, we assume that N ≥ 4. We construct a function u _aux ∈ S ^4,9 such that

$$u_{\text{aux}}^{(k)}(t_{i})=f_{i,k},\qquad0\leq k\leq4,~0\leq i\leq N, $$

where, for 0 ≤ i ≤ N and 0 ≤ k ≤ 4, the values f _{i, k} are divided differences

$$f_{i,k}:=\left\{ \begin{array}{ll} \left[t_{j-k/2},\ldots,t_{j+k/2}\right]f, & k\text{even},\\ \left[t_{j-\frac{k+1}{2}},\ldots,t_{j-1},t_{j+1},\ldots,t_{j+\frac{k+1}{2}}\right]f, & k\text{ odd} \end{array} \right. $$

which are centered at

$$ j:=j_{i,k}:=\left\{ \begin{array}{ll} \left\lceil \frac{k}{2}\right\rceil,& i<\left\lceil \frac{k}{2}\right\rceil,\\ N-\left\lceil \frac{k}{2}\right\rceil,& i>N-\left\lceil \frac{k}{2}\right\rceil,\\ i, & \text{ otherwise}. \end{array} \right. $$

(60)

As usual for Hermite interpolation, we initialize the divided differences by

$$\left[\underset{(k+1)\times}{\underbrace{t_{\ell},\ldots,t_{\ell}}}\right]f:=f_{\ell,k}\quad\text{ for }~\ell=i-1,i~\text{ and }~0\leq k\leq4 $$

and employ for the remaining differences the recursion

$$\left[\underset{m\text{-times}}{\underbrace{t_{i-1},\ldots t_{i-1}}},\underset{n\text{-times}}{\underbrace{t_{i},\ldots,t_{i}}}\right]f:=\frac{\left[\underset{(m-1) \text{-times}}{\underbrace{t_{i-1},{\ldots} t_{i-1}}},\underset{n\text{-times}}{\underbrace{t_{i},\ldots,t_{i}}}\right] f-\left[ \underset{m\text{-times}}{\underbrace{t_{i-1},{\ldots} t_{i-1}}},\underset{(n-1)\text{-times} }{\underbrace{t_{i},\ldots,t_{i}}}\right] f}{h_{i}} $$

for m ≥ 1 and n ≥ 1. On an interval τ = [t _i−1, t _i ], this leads to

$$u_{\text{aux}}|_{\tau_{i}}=\sum\limits_{k=0}^{4}f_{i-1,k}\omega_{i-1,k}+{\sum}_{k=0}^{4} \left( \left[\underset{5\times}{\underbrace{t_{i-1},\ldots,t_{i-1}}},\underset{(k+1) \times}{\underbrace{t_{i},\ldots,t_{i}}}\right]f\right)\omega_{i,k}, $$

where, for 0 ≤ k ≤ 4,

$$\omega_{\ell,k}(t) :=\left\{ \begin{array}{ll} \left( t-t_{i-1}\right)^{k}, & \ell=i-1,\\ \left( t-t_{i-1}\right)^{5}\left( t-t_{i}\right)^{k}, & \ell=i. \end{array} \right. $$

Thus, we get for 0 ≤ m ≤ 4

$$\left|\omega_{\ell,k}^{(m)}(t)\right| \leq C_{1}\left\{ \begin{array}{ll} 0, & m>k\wedge\ell=i-1,\\ h_{\tau_{i}}^{k-m}, & m\leq k\wedge\ell=i-1,\\ h_{\tau_{i}}^{5+k-m}, & \ell=i. \end{array} \right. $$

We introduce the index neighborhood (cf. (60))

$$\iota_{i}:=\left\{ j_{i-1,4}-2,\ldots,j_{i-1,4}+2\right\}\cup\left\{j_{i,4}-2,\ldots,j_{i,4}+2\right\} $$

which contains those indices which are involved in the definition of $\left [ \underset {m\text {-times}}{\underbrace {t_{i-1},\ldots t_{i-1}}},\underset {n\text {-times}}{\underbrace {t_{i},\ldots ,t_{i}}}\right ] f$ for 0 ≤ m ≤ 4, 0 ≤ n ≤ 4 with (m, n)≠(0,0). We set

$$h_{i,\min}:=\min\{h_{n}:n\in\iota_{i}\} $$

and obtain

$$\left[\underset{m\text{-times}}{\underbrace{t_{i-1},\ldots t_{i-1}}},\underset{n\text{-times}}{\underbrace{t_{i},\ldots,t_{i}}}\right] f\leq C_{2}\frac{\left\|(f_{n,k})_{n\in\iota_{i}}\right\|_{\max}}{h_{i,\min}^{m+n-k-1}}\quad\text{ for }~0\leq k\leq m+n-1\leq4. $$

Thus, for 0 ≤ r ≤ m ≤ 4, it holds

$$\begin{array}{@{}rcl@{}} \left\|u_{\text{aux}}^{(m)}\right\|_{C^{0}(\tau_{i})}&\leq & \sum\limits_{s=m}^{4}\frac{\left\|(f_{n,r})_{n\in\iota_{i}}\right\|}{h_{i,\min}^{s-r}}h_{\tau_{i}}^{s-m}+\sum\limits_{s=0}^{4}\frac{\left\|(f_{n,r})_{n\in\iota_{i}}\right\|}{h_{i,\min}^{s+5-r}}h_{\tau_{i}}^{s+5-m},\\ &\leq & C_{1}C_{2}\left( \sum\limits_{s=m}^{4}\left\|(f_{n,r})_{n\in\iota_{i}}\right\|_{\max}\left( \frac{h_{\tau_{i}}}{h_{i,\min}}\right)^{s-r}h_{\tau_{i}}^{r-m}\right. \\ && \left. +\sum\limits_{s=0}^{4}\left\|(f_{n,r})_{n\in\iota_{i}}\right\|_{\max}\left( \frac{h_{\tau_{i}}}{h_{i,\min}}\right)^{5+s-r}h_{\tau_{i}}^{r-m}\right) \\ &\leq & C_{3}h_{\tau_{i}}^{r-m}\left\|(f_{n,r})_{n\in\iota_{i}}\right\|_{\max}, \end{array} $$

(61)

where C ₃ only depends on the local quasi uniformity, i.e., on κ. This leads to

$$\|u_{\Delta}\|\leq \left\|u_{\text{aux}}\right\| + \left\| u_{\Delta}-u_{\text{aux}}\right\| \leq C_{3}\|\mathbf{f}\|_{\max}+\left\| u_{\Delta }-u_{\text{aux}}\right\|. $$

Proof of Case a. We employ [12,Theorem 2] and (61) to obtain

$$ \left\|u_{\Delta}-u_{\text{aux}}\right\| \leq Ch^{4}\left\|u_{\text{aux}}^{(4)}\right\| \leq C\left( \frac{h}{h_{\min}}\right)^{4}\|\mathbf{f}\|_{\max}. $$

(62)

To prove (56), we assume f ∈ C ^ℓ(I) for some 0 ≤ ℓ ≤ 4. Then, well known properties of divided differences imply

$$ \max_{1\leq i\leq N}\left\{h_{\tau_{i}}^{\ell-m}\left\|\left( f_{n,\ell}\right)_{n\in\iota_{i}}\right\|_{\max}\right\} \leq Ch^{\ell-m}\left\|f^{(\ell)}\right\| $$

(63)

and similarly as in (62) we get

$$\left\|u_{\Delta}-u_{\text{aux}}\right\| \leq Ch^{4}\left\|u_{\text{aux}}^{(4)}\right\| \leq Ch^{\ell}\left\|f^{(\ell)}\right\|, $$

where C depends on the global shape regularity h/h _min.

Proof of Case b. From [12]Corollary, we conclude that on an interval τ = [t _i−1, t _i ] it holds

$$ \left\|u_{\Delta}-u_{\text{aux}}\right\|_{C^{0}(\tau)}\leq\frac{1}{8}h_{\tau}^{2}\left( \frac{h_{\tau}^{2}}{8}\left\|u_{\text{aux}}^{(4)}\right\| _{C^{0}(\tau)}+\max_{\ell\in\{i-1,i\}}\left\{\left| u_{\text{aux}}^{(2)}\left( t_{\ell}\right) -u_{\Delta}^{(2)}\left( t_{\ell}\right)\right|\right\}\right). $$

(64)

The first term in the right-hand side of (64) can be estimated by using (61) with m = 4

$$ \left( \frac{h_{\tau}^{2}}{8}\right)^{2}\left\| u_{\text{aux}}^{(4)}\right\|_{C^{0}(\tau)}\leq C_{4}\left\|\left( f_{n}\right)_{n\in\iota_{i}}\right\|_{\max}, $$

(65)

where C ₄, again, only depends on the local quasi uniformity, i.e., on κ. Let ℓ∈{1,…,N−1} be such that

$$h_{\ell}^{2}\left|d_{\ell}^{(2)}\right| =\left\|\mathbf{h}^{2}\mathbf{d}^{(2)}\right\|_{\max} $$

with

$$d_{i}^{(2)}:=u_{\text{aux}}^{(2)}\left( t_{i}\right) - u_{\Delta}^{(2)}\left( t_{i}\right),\quad \mathbf{h}^{2}\mathbf{d}^{(2)}:=\left( {h_{i}^{2}}d_{i}^{(2)}\right)_{i=1}^{N-1},\quad h_{i}:=\frac{h_{\tau_{i}}+h_{\tau_{i+1}}}{2}. $$

Then, according to [12](2), the differences $d_{i}^{(2)}$ satisfy the relation

$$ \alpha_{\ell}d_{\ell-1}^{(2)}+2d_{\ell}^{(2)}+\left( 1-\alpha_{\ell}\right)d_{\ell+1}^{(2)}=r_{\ell} $$

(66)

with $\alpha _{\ell }=\frac {h_{\tau _{\ell }}}{h_{\tau _{\ell }}+h_{\tau _{\ell +1}}}$ and

$$r_{\ell}:=\alpha_{\ell}u_{\text{aux}}^{(2)}\left( t_{\ell-1}\right)+2u_{\text{aux}}^{(2)}\left( t_{\ell}\right) +\left( 1-\alpha_{\ell}\right)u_{\text{aux}}^{(2)}\left( t_{\ell+1}\right) -6\left[t_{\ell-1},t_{\ell},t_{\ell+1}\right] f. $$

Multiplying (66) by $h_{\ell }^{2}$ leads to⁴

$$h_{\ell}^{2}r_{\ell} =\alpha_{\ell}\left( \frac{h_{\ell}}{h_{\ell-1}}\right)^{2}h_{\ell-1}^{2}d_{\ell-1}^{(2)}+2h_{\ell}^{2}d_{\ell}^{(2)}+\left( 1-\alpha_{\ell}\right) \left( \frac{h_{\ell}}{h_{\ell+1}}\right)^{2}h_{\ell+1}^{2}d_{\ell+1}^{(2)}\geq h_{\ell}^{2}d_{\ell}^{(2)}\left( 2-\kappa^{2}\right). $$

Hence

$$\left\|\mathbf{h}^{2}\mathbf{d}^{(2)}\right\|_{\max}\leq\frac{1}{2-\kappa^{2}}\left\| \mathbf{h}^{2}\mathbf{r}\right\|_{\max}. $$

From [12](2), we conclude that

$$\left| {h_{i}^{2}}r_{i}\right| \leq2{h_{i}^{4}}\left\|u_{\text{aux}}^{(4)}\right\|_{C^{0}\left( \omega_{i}\right)}\overset{(61)}{\leq}C_{5}\left\|\left( f_{n}\right)_{n\in\iota_{i}^{\star}}\right\|_{\max} $$

with

$$\omega_{i}:=\bigcup\{\tau\in{\Delta}\mid t_{i}\in\tau\}\quad\text{ and }\quad\iota_{i}^{\star}:=\bigcup_{k:t_{i}\in\tau_{k}}\iota_{k}. $$

Thus, the second term in (64) can be estimated by

$$\frac{1}{8}h_{\tau}^{2}\max_{\ell\in\{i-1,i\}}\left\{\left| u_{\text{aux}}^{(2)}\left( t_{\ell}\right)-u_{\Delta}^{(2)}\left( t_{\ell}\right)\right|\right\} \leq C_{6}\|\mathbf{f}\|_{\max} $$

and (58) follows.

To prove (59), we assume f ∈ C ^ℓ(I) for some 0 ≤ ℓ ≤ 4 and estimate next the first term in the right-hand side in (65) by using (61)

$$\left( \frac{h_{\tau}^{2}}{8}\right)^{2}\left\| u_{\text{aux}}^{(4)}\right\|_{C^{0}(\tau)}\leq C_{7}h^{\ell}\left\| f^{(\ell) }\right\|. $$

For the second term in (64), we first observe that

$$\max_{1\leq i\leq N-1}\left| {h_{i}^{2}}r_{i}\right| \leq2\max_{1\leq i\leq N-1}{h_{i}^{4}}\left\|u_{\text{aux}}^{(4)}\right\|_{C^{0}\left( \omega_{i}\right)}\overset{(61)}{\leq}Ch^{\ell}\left\| f^{(\ell) }\right\| $$

holds so that the second term in (64) can be estimated by

$$\frac{1}{8}h_{\tau}^{2}\max_{\ell\in\{i-1,i\}}\left\{\left| u_{\text{aux}}^{(2)}\left( t_{\ell}\right) -u_{\Delta}^{(2)}\left( t_{\ell}\right) \right|\right\} \leq Ch^{\ell}\left\| f^{(\ell)}\right\|, $$

where C only depends on the local quasi-uniformity of the mesh and (58) follows. □