A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations (European Journal of Pure and Applied Mathematics, 2024)
In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation(WSVIE), $u(x)=f(x)+\int_{0}^{x}\frac{t^{μ-1}}{x^μ}[u(t)]^β dt$. AL-Jawary and Shehan used Daftardar-Jafari Method(DJM) and solved the above integral equation for the investigation parameter $μ>1$ using specific force functions with $μ$ and $β$ values and obtained unique solutions. We have discovered a force function $f(x)=x^{k_1}-{\frac{x^{γ k_1}}{γ k_1+μ}}$, that allows the introduction of noise terms phenomena discovered by Wazwaz; that cancel out the terms of the power series in the successive solution terms $u_m$, $m=0,1,2,...,n$: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by $x^{g(n)}$. Such that the integral solution can be written as $u(x)=u_0+\sum_{m=1}^{n}u_m$, where $n$ is finite. Simplifying the solution terms we get the unique solution $u(x)=x^{k_1}$, irrespective of $n-$value in the truncation point. We discovered a formula relation between the last solution term $u_n$ and the truncation point as $u_n=a_nx^{g(n)}$. Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter $μ>1$. We extend the parameter range to include $μ>1$ and $00$ and for any finite value of $n≥2$.
Keywords: Volterra integral equations; Weakly singular kernel; Force function formula; Series solutions; Unique solution