Under some suitable conditions, we study the existence and uniqueness of a solution to a new modification of a nonlinear fractional integro-differential equation (NFIDEq) in dual Banach space C_E (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 (BMA) method as a numerical method. The major motivation for selecting the BMA approach is that it combines Bernoulli polynomial approximation with Caputo fractional derivatives and numerical integral transformation to reduce the NFIDEq 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.

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space $C_E\left(E, J\right)$. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters $\lambda$ and time $t$ as well as how the absolute error in each example is calculated.

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.

In this article, we describe the existence and uniqueness of a solution to the nonlinear fractional Volterra integro differential equation in complex space using the fixed-point theory. We also examine the remarkably effective Euler wavelet method, which converts the model to a matrix structure that lines up with a system of algebraic linear equations; this method then provides approximate solutions for the given problem. The proposed technique demonstrates superior accuracy in numerical solutions when compared to the Euler wavelet method. Although we provide two cases of computational methods using MATLAB R2022b, which could be the final step in confirming the theoretical investigation.

This article presents a novel approach to looking at steady-state thermoelastic boundary value problems in isotropic elastic plates with curvilinear holes using a complex variable approach and rational conformal mappings. The physical domain with a non-circular opening is mapped conformally to the unit disk. A thermoelastic potential combines the temperature distribution, which is determined by the Laplace equation with Neumann boundary conditions. Gaursat functions, which are shown as truncated power series, show the complicated stress and displacement fields. They are found by putting boundary constraints at certain collocation points. This procedure presents us with a linear system that can be solved using the least squares method. The method is applied in an annular shape that is exposed to a radial temperature gradient. This experiment shows how changes at the boundary affect the distribution of stress. According to numerical simulations, stress distributions are more uniform when boundaries are smoother, but stress concentrations increase with the size of geometric disturbances. The suggested approach remarkably captures the way geometry and thermal effects interact in two-dimensional thermoelasticity. It is a reliable tool for researching intricate, heated elastic domains.

Fractional differential equations are highly beneficial in economics because they can be used to analyze nonlinear systems with memory effects. This research investigates a group of nonlinear fractional Riccati equations that show up in models of inventory and economic growth. The present work is a combined semi-analytical method for finding deterministic series solutions in the Caputo sense: the Adomian Decomposition–Sumudu Transform Method. Limited studies have examined its usage in memory-affected economic models. This method is effective with nonlinearities due to its ability to operate without the necessity of linearizing or discretizing them. The Mittag-Leffler function is employed to demonstrate that the series converges in a strict manner if it converges in a manner that is both absolute and uniform when the conditions are met. Finally, a Lyapunov stability study is conducted to ensure that the solution can accommodate modifications to the original data. Numerical models with different fractional orders show that the behavior of the system is controlled by the fractional parameter. When the fractional order is small, memory effects are increasing. As the order approaches closer to one, the solutions start to act like classical ones. These results indicate that the current methodology can be used for practical applications such as short-term currency exchange rates and volatility in financial markets.