|
چکیده
|
In this thesis, a piecewise-linear collocation method is developed together with a comparative analysis against the hat functions approximation technique for solving locally loaded Volterra integral equations. The presence of discrete load terms induces structural discontinuities in the solution and poses significant challenges both in theoretical analysis and numerical approximation. First, the existence and uniqueness of continuous solutions to the linear locally loaded Volterra integral equation are established under appropriate continuity assumptions imposed on the kernel, source function, and load coefficients, together with the nonsingularity condition of the associated matrix. Subsequently, a robust piecewise-linear collocation scheme is formulated in which the solution interval is partitioned into subintervals, the unknown solution is approximated by continuous piecewise-linear basis functions, and the load values are incorporated as auxiliary unknowns within a unified linear algebraic system. Following the solution of this system and the generation of the approximate solution via the proposed method, second-order accuracy is rigorously proven in both the L2 and L norms under sufficient smoothness conditions through convergence analysis. Finally, the theoretical convergence rates are verified through numerical examples presented in the concluding chapter.
|