A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

Abstract

In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation(WSVIE),  $u(x)=f(x)+{\int }_0^x\frac{t^{\mu -1}}{x^{\mu }}[u(t){]}^{\beta }dt$. AL-Jawary and Shehan used Daftardar-Jafari Method(DJM) and solved the above integral equation for the investigation parameter $\mu >1$ using specific force functions with $\mu$ and $\beta$ values and obtained unique solutions. We have discovered a force function $f(x)=x^{k_1}-\frac{x^{\gamma k_1}}{\gamma k_1+\mu }$, 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 $u_n=a_n{x}^{g(n)}$. Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter $\mu >1$. We extend the parameter range to include $\mu >1$ and $0<\mu \le 1$ for our solution. In addition, for any chosen rational parameter $k_1$, the solution $u(x)=x^{k_1}$ is extrapolated to be valid for all integer parameter values $\beta \ge 2$, and positive rational parameter value $\mu >0$  and for any finite value of $n\ge 2$.