An optimal control problem associated to a class of fractional Burgers’ equations

Mohammadizadeh, Fatemeh; Tehrani, Hojjat Ahsani; Georgiev, Svetlin G.; Skandari, Mohammad Hadi Noori

doi:10.1142/s1793557120500795

Cited by 4 publications

(2 citation statements)

References 11 publications

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“…For dealing with FPDEs, the classical CSC method is updated by using a new set of basis functions for interpolating the time variable. One of the most important advantages of FCSC method compared with those of other approximate methods is the high degree of accuracy that FCSC approximations offer for solving fractional calculus problems 17,34,35 …”

Section: Space–time Fractional Chebyshev Spectral Collocation Methodsmentioning

confidence: 99%

Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

Zadeh

Skandari

et al. 2020

Math Methods in App Sciences

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This article contributes to a balanced space–time spectral collocation method for solving nonlinear time‐fractional Burgers equations with given initial‐boundary conditions. Most of existing approximate methods for solving partial differential equations are unbalanced, since they have used a low order scheme such as finite difference methods for integrating the temporal variable and a high order numerical framework such as spectral Galerkin (or meshless) method for discretization of space variables. So in the current paper, our suggested scheme is balanced in both time and space variables. Due to the non‐smoothness of solutions of time‐fractional Burgers equations, we apply efficient basis functions as the fractional Lagrange functions for interpolating time variable. By collocating the main equation and the initial‐boundary conditions together with the implementation of the corresponding operational matrices of spatial and fractional temporal variables, the assumed model is transformed into the associated system of nonlinear algebraic equations, which can be solved via efficient iterative solvers such as the Levenberg–Marquardt method. Also, we fully analyze the convergence of method. Moreover, we consider several test problems for examining the suggested scheme that confirms its high accuracy and low computational cost with respect to recent numerical methods in the literature.

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Section: Space–time Fractional Chebyshev Spectral Collocation Methodsmentioning

confidence: 99%

Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

Zadeh

Skandari

et al. 2020

Math Methods in App Sciences

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“…Momani [24] has obtained the non-perturbative analytical solutions of space-and time-fractional Burgers' equations using Adomian decomposition method and describe the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. Mohammadizadeh et al [23] have obtained unique solutions under some special conditions for a special class of fractional Burgers' equation. They have also found at least one optimal solution for this problem.…”

Section: Introductionmentioning

confidence: 99%

Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation

Mittal

2020

Arab. J. Math.

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In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

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Numerical solution of ψ-Hilfer fractional Black–Scholes equations via space–time spectral collocation method

Mohammadizadeh¹,

Georgiev²,

Rozza³

et al. 2023

Alexandria Engineering Journal

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An optimal control problem associated to a class of fractional Burgers’ equations

Abstract: In this paper, we prove that a class of fractional Burgers’ equations has a unique solution under some special conditions. Moreover, we show that an optimal control problem for a special class of fractional Burgers’ equations has at least one optimal solution.

Cited by 4 publications

References 11 publications

Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

Space–time Chebyshev spectral collocation method for nonlinear time‐fractional Burgers equations based on efficient basis functions

Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation

Numerical solution of ψ-Hilfer fractional Black–Scholes equations via space–time spectral collocation method

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