Higher-order Haar wavelet method for the numerical solution of integro-differential equations

Purpose In this paper, we use the higher-order Haar wavelet method (HOHWM) for the approximate solution of first-order integro-differential equations (IDEs) of the second-kind. HOHWM is an improvement of the most popular HWM and depends on a parameter $\lambda$. The method is applied to both Volterra and Fredholm types of IDEs. The advantages of this method include higher accuracy, ease of implementation and good computing efficiency. The method is applied to several problems available in the literature. In the case of HOHWM, second- and fourth-order convergence is observed, which is an improvement over the first-order convergence of the HWM. Design/methodology/approach Assume that the equation has the highest order $n$ derivative. Next \begin{equation}\label{2} \frac{dˆ{n+2\lambda}w(t)}{dtˆ{n+2\lambda}} = \sum_{i = 1}ˆ{\infty}a_ih_i(t), \end{equation} $\lambda = 1,2,\dots$ in this case, {$h_i$,s are Haar functions having three values $0$, $1$ and $-1$}. Because of symmetry and more precise results, we use the even increment $2\lambda$ \cite{majakcs2018}. Values for the unknown function and its derivatives are found by integrating the previous formula. Integrating $n+2\lambda$ times Eq. \eqref{2} {yields the following formula:}\begin{equation}\label{3} w(t) = \frac{a_1tˆ{n+2\lambda}}{(n+2\lambda)!}+\sum_{k = 0}ˆ{\infty}\sum_{l = 0}ˆ{2ˆk-1}a_{2ˆk + l+1}Q_{n+2\lambda,2ˆk + l+1}(t) + S(t) + H(t), \end{equation} \(t = 2ˆk\), \(l = 0, \dots, t-1\), and \(T = 2ˆK\). Here, $t$ denotes the Haar wavelet (HW) resolution and $T$ denotes the greatest degree of resolution. Definitions of \(S(t)\) and \(H(t)\) in Eq.\eqref{3} are: \begin{equation} S(t) = \sum_{r = 0}ˆ{n-1}c_r\frac{tˆr}{r!},\qquad H(t) = \sum_{r = n}ˆ{n+2\lambda-1}c_r\frac{tˆr}{r!}. \end{equation} Findings The aforementioned process involves the $n+2\lambda$ variables $c_r,r = 0,\dots,n+2\lambda-1$. The original equation can be used to determine the values of the remaining $2\lambda$ constants, but the BCs can be used to calculate the values of the $n$ integration constants. Specifically, we choose $2\lambda$ points from the domain and insert them into discretized equations to generate $2\lambda$ more equations. These $2\lambda$ points and the collocation points ought to be distinct. There are countless ways to determine the remaining $2\lambda$ constants since there are countless ways to take the $2\lambda$ constants. Originality/value Numerical solutions of boundary-value problems of IDEs and differential equations are considered. The higher-order HW collocation method is applied. The method works equally well for linear as well as nonlinear problems. The method is applied to several test problems. Fourth-order convergence is obtained.