Reproducing kernel method for integro-differential equation with Abel kernel (Q2885019)

The paper is concerned with the solution of a linear integro-differential equation with a singular kernel of the form NEWLINE\[NEWLINE(Au)(t)+\lambda \int_a^t \frac{1}{(t-s)^{1/2}}(Lu)(s)ds=f(t), ~a<t<b,NEWLINE\]NEWLINE where \(L\) and \(A\) are linear operators and \(\lambda\) is a given number. The authors consider the equation in the form \(Au+Ku=f\) for suitably defined \(K\). The exact solution is represented by an infinite series in the reproducing kernel space \(W_2^m[a,b]\) whilst an approximate solution is obtained in the sense of the norm of the same reproducing kernel space.NEWLINENEWLINEExisting methods for solving this equation are listed in the introduction. Section 2 includes definitions of the reproducing kernel space (along with the relevant inner product and norm) and an existence theorem stating that the Hilbert space \(W_2^m[a,b]\) is a reproducing kernel space. Section 3 includes several lemmae and their proofs, a theorem proving that a unique solution exists if the kernel is a \(L_2\)-Volterra kernel and a theorem stating that the map \(A+K\) is an injection.NEWLINENEWLINEWith \(\psi_i(x)=(A_yR^m(x,y))(x_i)+(K_yR^m(x,y))(x_i)\), Section 4 includes the proof of a lemma that states that \(\psi_i(x)\in W_2^m[a,b]\) and the proof of a theorem stating that \(\{\psi_i\}_{i=1}^\infty\) is a complete system in \(W_2^m[a,b]\). Theorem 4.2 proves a form for the exact solution of the equation and in Corollary 4.1 uniform convergence is proved for an approximate solution.NEWLINENEWLINEThe authors state that advantages of the reproducing kernel method (RKM) are a fast speed of convergence, high precision and an improvement in dealing with the singularity. In Section 5, the efficiency of the RKM is illustrated by two examples.