A new type of equation of motion and numerical method for a harmonic oscillator with left and right fractional derivatives. (English) Zbl 07848635
Summary: The aim of this research is to propose a new fractional Euler-Lagrange equation for a harmonic oscillator. The theoretical analysis is given in order to derive the equation of motion in a fractional framework. The new equation has a complicated structure involving the left and right fractional derivatives of Caputo-Fabrizio type, so a new numerical method is developed in order to solve the above-mentioned equation effectively. As a result, we can see different asymptotic behaviors according to the flexibility provided by the use of the fractional calculus approach, a fact which may be invisible when we use the classical Lagrangian technique. This capability helps us to better understand the complex dynamics associated with natural phenomena.
MSC:
| 26Axx | Functions of one variable |
| 34Axx | General theory for ordinary differential equations |
| 35Rxx | Miscellaneous topics in partial differential equations |
Keywords:
fractional calculus; Euler-Lagrange equation; Caputo-Fabrizio derivative; harmonic oscillator; position-dependent massReferences:
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