Abstract:

The fractional of Cauchy-type problem (FCT) is solved precisely in the current study utilizing the Laplace residual power series approach (LRPS). The Caputo operator is used to determine the fractional derivative. Firstly, we present a brand-new technique that combines the residual power series strategy with the Laplace transform technique. We provide precise instructions for employing the suggested methodology to calculate fractional Cauchy-type formula. Next, we assess the technique’s effectiveness and accuracy using the FCT. The calculated and actual results are examined using graphic representations of the results, demonstrating how much more accurate the proposed approach is. The table illustrates the findings for fractional approximations results for different fractional orders in addition to nonfractional approximations and correct results. It is shown that like the number of phrases inside the serial that solve the issues rises, the relation between the generated answers as well as the real solutions to every issue converges. To exemplify that how proposed scheme works in calculating various types of fractional ordinary differential equations, two instances are provided.

Keywords: Fractional power series, Fractional Cauchy-type formula, Series of Laplace residual power, Laurent’s series.