A Legendre wavelet operational matrix method (LWM) is presented for the solution of nonlinear fractional-order Riccati differential equations, having variety of applications in quantum chemistry and quantum mechanics. The fractional-order Riccati differential equations converted into a system of algebraic equations using Legendre wavelet operational matrix. Solutions given by the proposed scheme are more accurate and reliable and they are compared with recently developed numerical, analytical, and stochastic approaches. Comparison shows that the proposed LWM approach has a greater performance and less computational effort for getting accurate solutions. Further existence and uniqueness of the proposed problem are given and moreover the condition of convergence is verified. 1. Introduction In recent years, use of fractional-order derivative goes very strongly in engineering and life sciences and also in other areas of science. One of the best advantages of use of fractional differential equation is modeling and control of many dynamic systems. Fractional-order derivatives are used in fruitful way to model many remarkable developments in those areas of science such as quantum chemistry, quantum mechanics, damping laws, rheology, and diffusion processes [1–5] described through the models of fractional differential equations (FDEs). Modeling of a physical phenomenon depends on two parameters such as the time instant and the prior time history; because of this reason, reasonable modeling through fractional calculus was successfully achieved. The abovementioned advantages and applications of FDEs attracted researchers to develop efficient methods to solve FDEs in order to get accurate solutions to such problems and more active research is still going on in those areas. Most of the FDEs are complicated in their structure; hence finding exact solutions for them cannot be simple. Therefore, one can approach the best accurate solution of FDEs through analytical and numerical methods. Designing accurate or best solution to FDEs, many methods are developed in recent years; each method has its own advantages and limitations. This paper aims to solve a FDE called fractional-order Riccati differential equation, one of the important equations in the family of FDEs. The Riccati equations play an important role in engineering and applied science [6], especially in quantum mechanics [7] and quantum chemistry [8, 9]. Therefore, solutions to the Riccati differential equations are important to scientists and engineers. Solving fractional-order Riccati differential equation,
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