Introduction to local fractional derivative and integral operators

We present the recent introduction to fractional derivatives. We give the properties and theorems of local fractional derivatives and local fractional integrals. Specially, we focus upon the local fractional Rolle’s theorem, mean value theorem, Newton–Leibniz formula, integration by parts, and Taylor’s theorem within the local fractional integrals. We also present the heat equation, wave equation, Laplace equation, Klein–Gordon equation, Schrodinger equation, diffusion equation, transport equation, Poisson equation, linear Korteweg–de Vries equation, Tricomi equation, Fokker–Planck equation, Lighthill–Whitham–Richards equation, Helmholtz equation, damped wave equation, dissipative wave equation, Boussinesq equation, nonlinear wave equation, Burgers equation, forced Burgers equation, inviscid Burgers equation, nonlinear Korteweg–de Vries equation, modified Korteweg–de Vries equation, generalized Korteweg–de Vries equation, nonlinear Klein–Gordon equation, Maxwell’s equation, Navier–Stokes equation, and Euler’s equation in fractal dimensional space within the local fractional partial derivative operator. We utilize the Cantor-type cycle coordinate system, Cantor-type cylindrical coordinate system, and Cantor-type spherical coordinate system to observe the wave equation, Laplace equation, Poisson equation, Helmholtz equation, heat-conduction equation, damped wave equation, dissipative wave equation, diffusion equation, and Maxwell’s equations in fractal dimensional space. We also discuss the wave equation, heat-conduction equation, damped wave equation, and diffusion equation in the Cantor-type cylindrical symmetry form and in the Cantor-type spherical symmetry form.

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