Nonlinear fractional order boundary-value problems with multiple solutions


It is well-known that discovering and then calculating all branches of solutions of fractional order nonlinear differential equations with boundary conditions can be difficult even by numerical methods. To overcome this difficulty, in this chapter two semianalytic methods are presented to predict and obtain multiple solutions of nonlinear boundary value problems. These methods are based on the homotopy analysis method (HAM) and Picard method namely, predictor HAM and controlled Picard method. The used techniques are capable of predicting and calculating all branches of the solutions simultaneously. Four problems are solved, three of them are practical problems which are generalized in fractional order domain to show the efficiency and importance of these methods. And the solutions are calculated by simple procedures without any need for special transformations or perturbation techniques. © 2018 Elsevier Inc. All rights reserved.


Semary M.S., Hassan H.N., Radwan A.G.


Bratu’s problem; Caputo fractional derivative; H-curvenonlinear boundary value problems; Homotopy analysis method; Multiple solutions; Picard method; Positive solutions


Mathematical Techniques of Fractional Order Systems, PP. 37 to 74, Doi: 10.1016/B978-0-12-813592-1.00002-7

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Book Chapter

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