We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
A force function formula for solutions of nonlinear weakly singular Volterra integral equations (WSVIE).
- Authors
Sarfo, Kwasi Frempong; Denteh, William Obeng; Takyi, Ishmael; Darkwah, Kwaku Forkuoh
- Abstract
In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation (WSVIE), u(x) = f(x)+∫0x t μ-1/xµ [u(t)]βdt. Al-Jawary and Shehan used the 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 k1-x γk1/γk1+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 um, 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) = u0+∑ m n=1 um, where n is finite. Simplifying the solution terms, we get the unique solution u(x) = x k1, irrespective of the n−value in the truncation point. We have discovered a formula relation between the last solution term un and the truncation point as un = anx 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 0 < µ ≤ 1 for our solution. In addition, for any chosen rational parameter k1, the solution u(x) = x k1 is extrapolated to be valid for all integer parameter values β ≥ 2 and positive rational parameter values µ > 0 and for any finite value of n ≥ 2.
- Subjects
VOLTERRA equations; SINGULAR integrals; POWER series; INTEGRAL equations; NONLINEAR operators
- Publication
European Journal of Pure & Applied Mathematics, 2024, Vol 17, Issue 2, p1046
- ISSN
1307-5543
- Publication type
Article
- DOI
10.29020/nybg.ejpam.v17i2.5071