## 7 Examples
The kernel \(K(x,t)=e^{xt}\) is many times continuously differentiable on \([0,1]\times[0,1]\), but it is not separable. The input function \(f(x)\) is continuous on \([0,1]\). We apply several approximating schemes. DKM: Taylor series: We formulate the given integral equation as in (19). Specifically, we replace the kernel \(K(x,t)\) and the input function \(f(x)\) by finite segments of Taylor series of degree \(n\) (\(r=n\)) about \(0\) in \(t\) and \(x\), respectively, as follows  
\[K_{n}(x,t)=\sum_{j=1}^{n+1}\frac{x^{j-1}}{(j-1)!}t^{j-1},\quad f_{n}(x)=\sum_{j =1}^{n+1}\frac{d^{j-1}f(x)}{dx^{j-1}}\bigg{|}_{x=0}x^{j-1},\]  
and solve via (5) and (6). For \(\lambda_{A}=2\), and \(n=2\), we get  
\[\tilde{u}_{2}(x) = -\frac{2145e-7257}{1559}-\frac{(5184e-24420)x}{10913}-\frac{(4530 e-17550)x^{2}}{10913}.\]  
Analogous solutions are obtained for other larger values of \(n\); e.g., for \(n=4\) and \(n=8\), we have  
\[\tilde{u}_{4}(x) = 0.997193+0.998193x+0.499304x^{2}+0.166475x^{3}+0.041625x^{4},\] \[\tilde{u}_{8}(x) = 0.999999+0.99999x+0.499999x^{2}+0.166666x^{3}+0.041666x^{4}\] \[+0.008333x^{5}+0.001388x^{6}+0.000198x^{7}+0.000024x^{8}.\]  
Comparison of these approximate solutions with the exact solution  
\[u(x)=e^{x}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}+\frac{x^{5}}{ 120}+\frac{x^{6}}{720}+\frac{x^{7}}{5040}+\frac{x^{8}}{40320}+\ldots\]  
shows the excellent agreement accomplished even with small values of \(n\). Further, the maximum errors between the approximate solutions for various values of \(n\) and the exact solution are given in Table 3. The maximum error is located at the point \(x=1\) in all cases. The results are distinguished for their high accuracy. DKM: Polynomials: We follow the procedure in Section 4.2 and approximate \(K(x,t)\) and \(f(x)\) with interpolating polynomials of degree \(n\) (\(r=n\)). Expressing the integral equation in (48) in the form (28) and solving by means of Theorem 1 for \(\lambda_{A}=2\), we find  
\[\tilde{u}_{2}(x) = 1.003217+0.877792x+0.847809x^{2},\] Figure 1:0 Figure 1:1 Figure 1:2  
Table 4 shows the maximum errors between these approximate solutions and the exact solution and the points where they occur. As expected, the results are very accurate. Compared with those obtained by the Taylor series above, they are superior. \begin{table}
\begin{tabular}{c c c c c} \hline \(n\) & \(\mathbf{\epsilon_{n}}=\|\mathbf{u}-\tilde{\mathbf{u}}_{n}\|_{\infty}\) & \(\mathbf{\eta}=\frac{\xi_{n}}{|\mathbf{u}|}\) & \(\mathbf{100\%\times\eta}\) & _Ratio_ \\ \hline
2 & \(3.77\times 10^{-1}\) & \(1.39\times 10^{-1}\) & \(13.88\%\) & \\
4 & \(1.55\times 10^{-2}\) & \(5.70\times 10^{-3}\) & \(0.57\%\) & \(24.3\) \\
8 & \(4.14\times 10^{-6}\) & \(1.52\times 10^{-6}\) & \(1.52\times 10^{-4}\%\) & \(3.7\times 10^{3}\) \\
16 & \(3.62\times 10^{-15}\) & \(1.33\times 10^{-15}\) & \(1.33\times 10^{-13}\%\) & \(1.1\times 10^{9}\) \\ \hline \end{tabular}
\end{table} 
Table 3: IE (48), \(\lambda_{A}=2\). DKM: Taylor series. QM: Trapezoidal rule and Taylor series: Let us use the composite trapezoidal rule  
\[\int_{a}^{b}y(x)dx\approx\frac{h}{2}\Bigg{\{}y(x_{1})+2\sum_{j=2}^{n}y(x_{j})+y(x _{n+1})\Bigg{\}}, \tag{49}\]  
to approximate the integral in (48). Following the procedure in Section 5, we may initially express (48) in the form (34). However, to construct an explicit solution of a type such as a polynomial throughout the interval \([a,b]\), we substitute the components of the vector \(G\) and the function \(f(x)\) with finite segments of the Taylor series of degree \(r=n\) in \(x\) about \(0\), as in (36).