This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontin- uous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational itera- tion method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effec- tive analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional com- plex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.

A Tutorial Review on Fractal Spacetime and Fractional Calculus

A new computational approach for solving nonlinear local fractional PDEs

Local fractional derivatives were investigated intensively during the last few years. The coupling method of Sumudu transform and local fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper. The presented method is applied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative. The obtained results are given to show the advantages.

/pdf/local-fractional-sumudu-transform-with-application-to-ivps-ey02fpr3g6.pdf

Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

We perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

/pdf/local-fractional-adomian-decomposition-and-function-5frmxrcp1c.pdf

Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators

Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell’s equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.

/pdf/maxwell-s-equations-on-cantor-sets-a-local-fractional-5asqaif2is.pdf

Maxwell’s Equations on Cantor Sets: A Local Fractional Approach

A reconstructive scheme for variational iteration method using the
 Yang-Laplace transform is proposed and developed with the Yang-Laplace
 transform. The identification of fractal Lagrange multiplier is investigated
 by the Yang-Laplace transform. The method is exemplified by a fractal heat
 conduction equation with local fractional derivative. The results developed
 are valid for a compact solution domain with high accuracy.

/pdf/reconstructive-schemes-for-variational-iteration-method-3nsb2fyqeu.pdf

Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem

To solve the problem of the low segmentation accuracy of lung nodule CT images using U-Net, an improved method for segmentation of lung nodules by U-Net was proposed. Initially, the dense network connection and sawtooth expanded convolution design was added to the feature extraction part, and a local residual design was adopted in the upsampling process. Finally, the effectiveness of the proposed algorithm was evaluated using the LIDC-IDRI lung nodule public dataset. The results showed that the improved algorithm had 7.03%, 14.05%, and 10.43% higher performance than the U-Net segmentation algorithm under the three loss functions of DC, MIOU, and SE, and the accuracy was 2.45% higher compared with that of U-Net. Thus, the proposed method had an effective network structure.

/pdf/lung-nodule-ct-image-segmentation-model-based-on-multiscale-ljfvjse8.pdf

Shan-Shan Kong

Papers

Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem

Lung Nodule CT Image Segmentation Model Based on Multiscale Dense Residual Neural Network