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In this paper, we enlarge the class of <inline-formula><math display="inline"><semantics><msup><mi>C</mi><mo>*</mo></msup></semantics></math></inline-formula>-algebra valued partial metric spaces as well as the class of <inline-formula><math display="inline"><semantics><msup><mi>C</mi><mo>*</mo></msup></semantics></math></inline-formula>-algebra valued <i>b</i>-metric spaces by introducing the class of <inline-formula><math display="inline"><semantics><msup><mi>C</mi><mo>*</mo></msup></semantics></math></inline-formula>-algebra valued partial <i>b</i>-metric spaces and utilize the same to prove our fixed point results. We furnish an example to highlight the utility of our main result. Finally, we apply our result in order to examine the existence and uniqueness of a solution for the system of Fredholm integral equations.