A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

In the current study, we proposed an operational matrix‐based spectral computational method coupled with the Picard technique and successfully employed to seek the solutions to a family of nonlinear evolution differential equations. However, the operational matrices are constructed assisted by monomials, and relevant theorems are available to authenticate the approach mathematically. The iterative form of residual vector for the problems understudy has been presented to seek some new results of a nonlinear family of evolutionary fractional problems. The coupling of spectral technique with the Picard method can easily tackle the high nonlinearities including quadratic, cubic, exponential, and hyperbolic, while it brings more accurate and efficient results than the published results. A worthy comparative study with exact solutions and existing techniques including numerical and analytical is being made, which shows an excellent level of accuracy for the problems under study. The authenticity of the proposed method is validated via convergence, error‐bound, and norms, while the set of graphs is illustrated for integer and noninteger orders. Hence, this unlocks the opportunities to deal with solutions of the weakly singular integral and lattice Toda equations and other complex nonlinear problems arising in applied sciences and may provide a better observation of physical mechanism.

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“…Firstly, we will apply the Picard‐iterative technique [34] and the problem understudy reduces to following iterative scheme (see Equation (14)), where s ∈ Z + is representing the iterations while the initial guess for the said iterative scheme can be chosen via expression (15):

\frac{\partial^{v_{1}}}{\partial t^{v_{1}}} u_{s} false(x, t false) = \frac{\partial^{2 v_{2}}}{\partial x^{2 v_{2}}} u_{s} false(x, t false) + \frac{\partial^{2 v_{3}}}{\partial y^{2 v_{3}}} u_{s} false(x, t false) + ϵ u_{s} false(x, t false) + F false(u_{s - 1} false(x, t false) false) + G false(x, t false),

u_{0} false(x, t false) = u false(x, 0 false) .

…”

Section: Pcpm For Higher‐dimensional Problemsmentioning

confidence: 99%

Linearized stable spectral method to analyze two‐dimensional nonlinear evolutionary and reaction‐diffusion models

Hamid

Usman

Haq

et al. 2020

Numerical Methods Partial

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The work is devoted to the development of a new spectral method based on higher dimensional orthogonal polynomials. Firstly, the concept of traditional Chelyshkov polynomials is extended for the function of more than one variable while definitions and theorems are presented with proofs. The operational matrices of derivative have been constructed assisted by defined higher‐order polynomials and used to the development of a spectral method. The method is further coupled with a Picard‐iterative scheme to tackle the high nonlinearities and termed as the Picard–Chelyshkov polynomial method (PCPM). The convergence and error‐bound have been analyzed through theorems and their proofs in order to prove the mathematical authenticity of the method. The PCPM is applied for some two‐dimensional unsteady nonlinear fractional partial differential equations and efficient results have been attained. In addition, it is evident from the comparative analysis with existing literature that the proposed method is fair enough in terms of accuracy, efficiency, and cost to deal with the problem in higher dimensions whilst could be further modified for other classes of dynamical problems.

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A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

Cited by 24 publications

References 64 publications

Higher‐order algorithms for stable solutions of fractional time‐dependent nonlinear telegraph equations in space

Higher‐order algorithms for stable solutions of fractional time‐dependent nonlinear telegraph equations in space

Hybrid fully spectral linearized scheme for time‐fractional evolutionary equations

Linearized stable spectral method to analyze two‐dimensional nonlinear evolutionary and reaction‐diffusion models

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