| Summary: | <p> Natural lower and upper solutions for initial value problems guarantees the interval of existence. However, coupled lower and upper solutions used as initial approximation in generalized iterative method are very useful since the iterates can be computed without any extra assumption. Generalized monotone method, along with the method of lower and upper solutions, has been used to develop the coupled lower and upper solutions on an extended interval for both scalar and system of Caputo fractional differential equations. This method yields linear convergence. Generalized quasilinearization method, along with the method of lower and upper solutions, was used to compute the coupled minimal and maximal solutions, if coupled lower and upper solutions existed for the scalar Caputo fractional differential equations. This method yielded quadratic convergence. Also, a mixed method of monotone method and quasilinearization method was developed to compute the coupled minimal and maximal solutions, if coupled lower and upper solutions existed, for the scalar Caputo fractional differential equations. This mixed method was used to compute the coupled lower and upper solutions on the desired interval, which yielded superlinear convergence. Numerical examples have been provided as an application of the analytical results.</p>
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