Local Fractional Laplace Variational Iteration Method for Fractal Vehicular Traffic Flow

In this article, we introduce the local fractional integral iterative method and the local fractional new iterative method for solving the local fractional differential equations. Also, we perform a comparison between the results obtained by these 2 local fractional methods with the results obtained by some other local fractional methods. The obtained results illustrate the significant features of the 2 methods that are both very effective and straightforward for solving the differential equations with local fractional derivative compared with the other local fractional methods.

“…In this section, we mention the notations, definitions, and some properties of the local fractional operators and fractional calculus. ()…”

Section: Mathematical Fundamentalsmentioning

confidence: 99%

“…Example Consider the local fractional wave equation():

\frac{\partial^{2 α}}{\partial t^{2 α}} u false(x, t false) - \frac{x^{2 α}}{normalΓ false(1 + 2 α false)} \frac{\partial^{2 α}}{\partial x^{2 α}} u false(x, t false) = 0, 0 < α \leq 1,

with the initial conditions

u false(x, 0 false) = \frac{x^{2 α}}{normalΓ false(1 + 2 α false)}, \frac{\partial^{α}}{\partial t^{α}} u false(x, 0 false) = 0 .

LFIIM. According to , the local fractional wave Equation 36 is equivalent to the local fractional integral equation:

u_{r + 1} false(x, t false) = \frac{x^{2 α}}{normalΓ false(1 + 2 α false)} +_{0} I_{t}^{false(α false)}_{0} I_{t}^{false(α false)} ([], \frac{x^{2 α}}{normalΓ false(1 + 2 α false)} . \frac{\partial^{2 α}}{\partial x^{2 α}} u_{r} false(x, t false)), r \geq 0 .

Therefore, from , we can obtain the following first few components of the local fractional integral iterative solution for (36):

\begin{matrix} u_{0} false(x, t false) & = \frac{x^{2 α}}{normalΓ false(1 + 2 α false)}, \end{matrix}

…”

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Mathematical Fundamentalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Self Cite

Analytical method to solve the local fractional vehicular traffic flow model

Singh

Kumar

Goswami

et al. 2021

This paper presents a new method to solve the local fractional partial differential equations (LFPDEs) describing fractal vehicular traffic flow. Firstly, the existence and uniqueness of solutions to LFPDEs were proved and then two schemes known as the basic method (BM) and modified local fractional variational iteration method (LFVIM) were developed to solve the local fractional PDEs. Multiple studies have been reported in the literature to solve these problems using iterative methods which are time‐consuming and prone to errors. For linear problems, basic method was found highly accurate and computationally sound. We derived a modified version of LFVIM to investigate and obtain the nondifferentiable solutions of linear, nonlinear, and nonhomogeneous LFPDEs arising in fractal vehicular traffic flow through illustrative examples. Study results show that both schemes are very effective and can be used successfully to solve fractal vehicular traffic flow problems.