Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach

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<abstract> <p>Under some suitable conditions, we study the existence and uniqueness of a solution to a new modification of a nonlinear fractional integro-differential equation (<bold>NFIDEq</bold>) in dual Banach space C<sub>E</sub> (E, [0, T]), which simulates several phenomena in mathematical physics, quantum mechanics, and other domains. The desired conclusions are demonstrated with the use of fixed-point theorems after applying the theory of fractional calculus. The validation of the provided strategy has been done by utilizing the Bernoulli matrix approach (<bold>BMA</bold>) method as a numerical method. The major motivation for selecting the <bold>BMA</bold> approach is that it combines Bernoulli polynomial approximation with Caputo fractional derivatives and numerical integral transformation to reduce the <bold>NFIDEq</bold> to an algebraic system and then derive the numerical solution; additionally, the convergence analysis indicated that the proposed strategy has more precision than other numerical methods. Finally, as a verification of the theoretical work, we apply two examples with numerical results by using [Matlab R2022b], illustrating the comparisons between the exact solutions and numerical solutions, as well as the absolute error in each case is computed.</p> </abstract>