Systems of IDE

Example 1

    Let us consider the linear system of integro-diferential equations $$ \begin{cases} \displaystyle{ \frac{d}{dx}y +xz +\int_{0}^{1}\exp(x+t)y(t) dt-(x+x^2)\int_{0}^{x}\sin(x-t)z(t)dt=g_1 }\\ \displaystyle{ x^2y -3\frac{d}{dx}z -x\int_{0}^{x}(2x+\sin(t))y(t) dt-\int_{0}^{1}(x-y)z(t)dt=g_2 }\\ y(0)=1,\quad z(0)=0 \end{cases}. $$ The approximate solution on Tau Toolbox:

The approximate tau solution achieves the machine precision:

Example 2

    Let us consider the linear system of integral equations $$ \begin{cases} \displaystyle y_1 - \int_{0}^{x}\left(\sin(x-t)-1\right)y_1(t)dt - \int_{0}^{x}\left(1-t\cos(x) \right)y_2(t)dt= g_1 \\ \displaystyle y_2 - \int_{0}^{x}y_1(t)dt - \int_{0}^{x}(x-t)y_2(t)dt= g_2 \end{cases}, $$ where $g_1 =-\frac{1}{2}(x-2)\sin(x) -x\cos(x) ^2+(\sin(x) +2)\cos(x) -1$ e $g_2 =-x+\sin(x) $. The next code presents the approximate solution by Tau Toolbox and the comparison with other autors:

The approximate tau solution achieves the machine precision: