scholarly journals Monte carlo estimation of the solution of fractional partial differential equations

2021 ◽  
Vol 24 (1) ◽  
pp. 278-306
Author(s):  
Vassili Kolokoltsov ◽  
Feng Lin ◽  
Aleksandar Mijatović
Keyword(s):  
Exact Solution ◽  
Monte Carlo ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rates Of Convergence ◽  
Probabilistic Interpretation ◽  
Fractional Partial Differential Equations ◽  
Fractional Pdes ◽  
Monte Carlo Approximation ◽  
Monte Carlo Estimation

Abstract The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

scholarly journals Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Sami Aljhani ◽  
Mohd Salmi Md Noorani ◽  
Khaled M. Saad ◽  
A. K. Alomari
Keyword(s):  
Exact Solution ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Accurate Solution ◽  
Caputo Fractional Derivatives ◽  
Average Residual ◽  
Homotopy Analysis

A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ℏ u , v -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.

scholarly journals Low-Variance Monte Carlo Simulation of Thermal Transport in Graphene

2012 ◽  
Author(s):  
Colin Landon ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Heat Transfer ◽  
Monte Carlo ◽  
Thermal Management ◽  
Numerical Solutions ◽  
Graphene Nanoribbons ◽  
Boltzmann Transport Equation ◽  
Approximate Solutions ◽  
Simulation Method ◽  
Boltzmann Transport ◽  
Kinetic Description

Due to its unique thermal properties, graphene has generated considerable interest in the context of thermal management applications. In order to correctly treat heat transfer in this material, while still reaching device-level length and time scales, a kinetic description, such as the Boltzmann transport equation, is typically required. We present a Monte Carlo method for obtaining numerical solutions of this description that dramatically outperforms traditional Monte Carlo approaches by simulating only the deviation from equilibrium. We validate the simulation method using an analytical solution of the Boltzmann equation for long graphene nanoribbons; we also use this result to characterize the error associated with previous approximate solutions of this problem.

scholarly journals Chebyshev Polynomials of Sixth Kind for Solving Nonlinear Fractional PDEs with Proportional Delay and Its Convergence Analysis

2022 ◽  
Vol 2022 ◽  
pp. 1-20
Author(s):  
Khadijeh Sadri ◽  
Hossein Aminikhah
Keyword(s):  
Chebyshev Polynomials ◽  
Climate Models ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Proportional Delay ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Residual Function ◽  
Fractional Pdes ◽  
The One

This work devotes to solving a class of delay fractional partial differential equations that arises in physical, biological, medical, and climate models. For this, a numerical scheme is implemented that applies operational matrices to convert the main problem into a system of algebraic equations; then, solving the resultant system leads to an approximate solution. The two-variable Chebyshev polynomials of the sixth kind, as basis functions in the proposed method, are constructed by the one-variable ones, and their operational matrices are derived. Error bounds of approximate solutions and their fractional and classical derivatives are computed. With the aid of these bounds, a bound for the residual function is estimated. Three illustrative examples demonstrate the simplicity and efficiency of the proposed method.

Numerical solutions of coupled nonlinear fractional KdV equations using He’s fractional calculus

2020 ◽  
pp. 2150023
Author(s):  
Dianchen Lu ◽  
Muhammad Suleman ◽  
Muhammad Ramzan ◽  
Jamshaid Ul Rahman
Keyword(s):  
Partial Differential Equations ◽  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Computational Time ◽  
Fractional Partial Differential Equations ◽  
Transform Method ◽  
Kdv Equations ◽  
Differential Transform

In this paper, we determine the application of the Fractional Elzaki Projected Differential Transform Method (FEPDTM) to develop new efficient approximate solutions of coupled nonlinear fractional KdV equations analytically and computationally. Numerical solutions are obtained, and some major characteristics demonstrate realistic reliance on fractional-order values. The basic tools, properties and approaches introduced in He’s fractional calculus are utilized to explain fractional derivatives. The consistency of FEPDTM and the reduction in computational time give FEPDTM extensive applicability. Furthermore, the calculations concerned in FEPDTM are too simple and straightforward. It is verified that FEPDTM is an influential and efficient technique to handle fractional partial differential equations. It is being observed that FEPDTM is more efficient than known analytical and computational methods.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

scholarly journals Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

2019 ◽  
Vol 55 (1) ◽  
pp. 184-210 ◽  
Author(s):  
Pierre Henry-Labordère ◽  
Nadia Oudjane ◽  
Xiaolu Tan ◽  
Nizar Touzi ◽  
Xavier Warin
Keyword(s):  
Monte Carlo ◽  
Branching Diffusion ◽  
Monte Carlo Approximation ◽  
Semilinear Pdes

ANALYTICAL APPROXIMATE SOLUTIONS OF (N + 1)-DIMENSIONAL FRACTAL HARRY DYM EQUATIONS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850094
Author(s):  
JIANSHE SUN
Keyword(s):  
Approximate Solutions ◽  
Travelling Wave ◽  
Cantor Sets ◽  
Travelling Wave Solutions ◽  
Fractional Partial Differential Equations ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractal Models ◽  
Differential Transform ◽  
Harry Dym Equation

The new fractal models of the [Formula: see text]-dimensional and [Formula: see text]-dimensional nonlinear local fractional Harry Dym equation (HDE) on Cantor sets are derived and the analytical approximate solutions of the above two new models are obtained by coupling the fractional complex transform via local fractional derivative (LFD) and local fractional reduced differential transform method (LFRDTM). Fractional complex transform for functions of [Formula: see text]-dimensional variables is generalized and the theorems of [Formula: see text]-dimensional LFRDTM are supplementary extended. The travelling wave solutions of the fractal HDE show that the proposed LFRDTM is effective and simple for obtaining approximate solutions of nonlinear local fractional partial differential equations.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

MONTE CARLO METHODS IN FUZZY GAME THEORY

2007 ◽  
Vol 03 (02) ◽  
pp. 259-269 ◽  
Author(s):  
AREEG ABDALLA ◽  
JAMES BUCKLEY
Keyword(s):  
Game Theory ◽  
Monte Carlo ◽  
Monte Carlo Method ◽  
Monte Carlo Methods ◽  
Mixed Strategy ◽  
Minimax Theorem ◽  
Approximate Solutions ◽  
Mixed Strategies ◽  
Fuzzy Game ◽  
Zero Sum

In this paper, we consider a two-person zero-sum game with fuzzy payoffs and fuzzy mixed strategies for both players. We define the fuzzy value of the game for both players [Formula: see text] and also define an optimal fuzzy mixed strategy for both players. We then employ our fuzzy Monte Carlo method to produce approximate solutions, to an example fuzzy game, for the fuzzy values [Formula: see text] for Player I and [Formula: see text] for Player II; and also approximate solutions for the optimal fuzzy mixed strategies for both players. We then look at [Formula: see text] and [Formula: see text] to see if there is a Minimax theorem [Formula: see text] for this fuzzy game.

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