article · Fractal and Fractional
A new computational approach, designated the Mittag-Leffler–Galerkin method, has been developed to numerically solve fractional-order Riccati differential equations. The technique combines the finite Mittag-Leffler function with the standard Galerkin framework to formulate approximate solutions. Mathematical analysis accompanies the method, delivering two distinct theorems that establish formal bounds on calculation errors. In addition, a residual correction method is introduced, serving to both estimate errors and refine calculations to yield improved approximate results. Demonstrations on various test problems confirm the practical execution and computational efficiency of the approach. Comparative evaluations indicate that the method achieves superior precision compared to certain established algorithms in existing literature, whilst producing results comparable to other recognised numerical procedures.
Fractional differential equations are complex mathematical models that often require specialised numerical techniques when exact solutions cannot be determined. By pairing the Mittag-Leffler function with the Galerkin method and incorporating formal error-bounding theorems, this work offers a reliable computational tool for solving fractional Riccati equations with accuracy that rivals or exceeds several conventional numerical approaches.
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We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods.
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DOI: 10.3390/fractalfract7040302
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