Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 11 2 2026 09 04 Convergence Quantum Analysis of Positive Solution for Caputo-Hadamard Fractional $q$-Differential Equations 204 219 2119 10.22128/ansne.2026.3296.1210 EN Mohammad Esmael Samei Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran 0000-0002-5450-3127 Somayeh Nazari Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran Journal Article 2026 04 13 In this research we obtain new results  of approximate solution for nonlinear singular $p$-Laplacian Caputo-Hadamard fractional $\mathtt{q}$-differential equation under infinite-point boundary conditions. The existence of unique iterative positive, error estimation, and convergence rate of approximate solution are obtained based on  the description of Green function with special properties and employing  appropriate substitution and appropriate cone. A few applications are demonstrated the validity of our achievements. https://ansne.du.ac.ir/article_2119_b0c81babd3acde4395ac30eaf59f7655.pdf