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Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations. (English) Zbl 1086.65121

The authors consider systems of $k$ linear integro-differential equations of Fredholm-Volterra type in the form

$\phantom{\rule{-0.166667em}{0ex}}\sum _{n=0}^{m}\sum _{j=1}^{k}{p}_{ij}^{n}\left(x\right){y}_{j}^{\left(n\right)}\left(x\right)\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{g}_{i}\left(x\right)\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}{\int }_{-1}^{1}\sum _{j=1}^{k}{F}_{ij}\left(x,t\right){y}_{j}\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}{\int }_{-1}^{x}\sum _{j=1}^{k}{K}_{ij}\left(x,t\right){y}_{j}\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt,\phantom{\rule{2.em}{0ex}}\left(1\right)$
$i=1,2,\cdots ,k,\phantom{\rule{1.em}{0ex}}-1⩽x⩽1,$

under the mixed conditions

$\sum _{n=0}^{m-1}{𝐚}_{j}^{n}{y}_{j}^{\left(n\right)}\left(-1\right)+{𝐛}_{j}^{n}{y}_{j}^{\left(n\right)}\left(1\right)+{𝐜}_{j}^{n}{y}_{j}^{\left(n\right)}\left(c\right)={\lambda }_{j},\phantom{\rule{1.em}{0ex}}j=1,2,\cdots ,k,\phantom{\rule{1.em}{0ex}}-1

where ${\lambda }_{j}$, ${𝐚}_{j}^{n}$, ${𝐛}_{j}^{n}$ and ${𝐜}_{j}^{n}$ are real-valued column matrices with $m×1$ dimension and ${y}_{j}^{\left(n\right)}$ indicates the $n$th-order derivative and ${y}_{j}^{\left(0\right)}\left(x\right)={y}_{j}\left(x\right)$. The aim of this study is to get a solution as truncated Chebyshev series defined by

${y}_{j}\left(x\right)=\sum _{r=0}^{N}{a}_{jr}{T}_{r}\left(x\right),\phantom{\rule{1.em}{0ex}}j=1,2,\cdots ,k,\phantom{\rule{1.em}{0ex}}-1⩽x⩽1,$

where ${T}_{r}\left(x\right)$ denotes the Chebyshev polynomials of the first kind, ${a}_{jr}$ are unknown Chebyshev coefficients, and $N$ is chosen any positive integer such that $N⩾m$.

The authors transform the system (1) and the given conditions (2) into matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. An interesting feature of this method is that when system of integro-differential equations (1) has linearly independent polynomial solution of degree $N$ or less than $N$, the method can be used for finding the analytical solution. Besides, when the truncation limit $N$ is increased, there exists a solution, which is closer to the exact solution. Some numerical results are also given to illustrate the efficiency of the method.

##### MSC:
 65R20 Integral equations (numerical methods) 45J05 Integro-ordinary differential equations 45F05 Systems of nonsingular linear integral equations