A Bessel collocation method for numerical solution of generalized pantograph equations (Q2910803)

This paper considers a generalized pantograph equation NEWLINE\[NEWLINE y^{(m)}(t) = \sum\limits_{j=0}^J \sum\limits_{k=0}^{m-1} P_{jk}(t) y^{(k)} (\lambda_{jk} t + \mu_{jk}) + g(t), \quad 0 \leq t \leq b, NEWLINE\]NEWLINE with initial condition NEWLINE\[NEWLINE \sum\limits_{k=0}^{m-1} c_{ij} y^{(k)}(0) = \lambda_i, \quad i=0,1,\dots,m-1, NEWLINE\]NEWLINE where \(P_{jk}(t)\) and \(g(t)\) are continuous functions in the interval \([0,b]\) and \(c_{ij}, \lambda_i, \lambda_{ij}\) and \(\mu_{jk}\) are real or complex constants. The authors propose to approximate the solution \(y(t)\) by a truncated series of Bessel polynomials. Using the collocation method with equidistant points, the expansion coefficients are found by solving a system of linear algebraic equations. They prove an \(L^\infty\) error estimate and present several numerical examples, where they compare the proposed method with spline, Adomian, Taylor and variational methods.