fractional

functional equation

fractional differential

fixed point theorems

fractional derivative

functional differential

common fixed

homotopy perturbation

fractional diffusion

Symmetry

The present paper aims to define three new notions: Θ e -contraction, a Hardy–Rogers-type Θ -contraction, and an interpolative Θ -contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.

Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space