Abstract
We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.
Highlights
Fractional calculus arises in many problems of physics, continuum mechanics, viscoelasticity, and quantum mechanics, and other branches of applied mathematics and nonlinear dynamics have been studied 1–7
We study the technique of the local fractional Fourier series for treating the local fractional wave equation in fractal vibrating string
We applied the technique of the local fractional Fourier series to treat with the local fractional wave equation in fractal vibrating string
Summary
Fractional calculus arises in many problems of physics, continuum mechanics, viscoelasticity, and quantum mechanics, and other branches of applied mathematics and nonlinear dynamics have been studied 1–7. We cannot employ the classical Fourier series, which requires that the defined functions should be differentiable, to describe some solutions to ordinary and partial differential equations in fractal space. Local fractional calculus is revealed as one of useful tools to deal with everywhere continuous but nowhere differentiable functions in areas ranging from fundamental science to engineering 42–57. For these merits, local fractional calculus was successfully applied in the local fractional Laplace problems 53, 54 , local fractional Fourier analysis 53, 54 , local fractional short time transform 53, 54 , local fractional wavelet transform 53–55 , fractal signal 55, , and local fractional variational calculus.
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