Expansion approach for solving nonlinear Volterra integro-differential equations (Q2878099)

The authors consider the following Volterra integro-differential equations of the form NEWLINE\[NEWLINED( x,y(x),y^{(1)}(x),\dots, y^{(\alpha)}(x)) - \lambda \int_{0}^{x} \varphi (x,t,y(t),y^{(1)}(t), \dots, y^{(\beta)}(t) ) dt= f(x),NEWLINE\]NEWLINE \(x,t \in \Gamma = [ 0,b],\) with the initial conditions NEWLINE\[NEWLINE\sum _{j=0} ^ {\alpha - 1 } B_{ij} y^{(j)} (0) = c_i, ~~ i = 1, 2, \dots , \alpha,NEWLINE\]NEWLINE where \(D\) and \(\varphi\) are in the following form NEWLINE\[NEWLINE\begin{multlined} D( x,y(x),y^{(1)}(x),\dots, y^{(\alpha)}(x)) \\ = \sum _{i=0} ^ {\mu_1} ( p_i (x) \prod_{j=0}^\alpha (y^{(j)}(x))^{\alpha_{ij}}), \varphi( x,t,y(t),y^{(1)}(t),\dots, y^{(\beta)}(t))\\ = \sum _{i=0} ^ {\mu_2} ( k_i (x,t) \prod_{j=0}^\beta (y^{(j)}(t))^{\beta_{ij}}),\end{multlined}NEWLINE\]NEWLINE and \(\alpha_{ij}, \beta_{ij} \in \mathbb N\cup \{0\}.\)NEWLINENEWLINEUsing the Taylor series expansion method, they consider the solution to be of the form NEWLINE\[NEWLINE y(x) = \sum _{j=0} ^ {N} e_{ij} x^j,NEWLINE\]NEWLINE where \( e_{ij} = \frac{y^{(j)}(0)}{j}\) are known from the given initial conditions for \( j = 0,1,\dots, a_{i}-1 \), and the remaining \(N-a_{i}\) values of \( e_{ij} \) are obtained by solving a system of \(N-a_{i}\) nonlinear equations which are obtained by substituting \(y(x)\) in the considered problem. These algebraic equations are solved using the powerful Grobrier basis method. The technique is clearly explained in Section 2 of the paper.NEWLINENEWLINEThe authors modify the Taylors series method from an expansion about \( n=0\) to an expansion about \( x=h \) and prove the necessary results so as to apply the Grobrier basis algorithm. Further, they study the error estimates and give an application to illustrate the performance and accuracy of the developed technique. This paper is quite interesting in its approach and technique.