A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions

Zogheib, Bashar; Tohidi, Emran

doi:10.1016/j.amc.2016.06.023

Cited by 15 publications

(21 citation statements)

References 36 publications

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“…By using these operational matrices and quadratures, computations time in numerical algorithms may be decreased significantly. Therefore, in most of the numerical methods, one of the above-mentioned tools may be applied for speeding up the operations in solving PDEs [5].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, in every considered PDE one may have some boundary conditions and we should impose these conditions in the procedure of the Tau matrix method. Our numerical implementation experiences show that if the boundary conditions are first imposed to the main equations [5] (via transforming the basic PDEs into the associated integro-PDEs), then we may have 2 Advances in Mathematical Physics more stable numerical solutions. However, in some research works (e.g., Taylor matrix method [6,7], Euler matrix method [8], and Chebyshev Tau method [9]) boundary conditions are imposed after discretizing the main equations.…”

Section: Introductionmentioning

confidence: 99%

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Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

Zogheib

Tohidi

Shateyi

2017

Advances in Mathematical Physics

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A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

Zogheib

Tohidi

Shateyi

2017

Advances in Mathematical Physics

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“…In spite of a lot of investigations on -Bernoulli polynomials, approximation properties of the -Bernoulli polynomials are not studied. We may apply these results to solve -difference equation, -analogue of the things that is done in [3] or [4]. The techniques of -operator can be applied in operator theory.…”

Section: Further Workmentioning

confidence: 99%

“…For example, at [1], some identities and formulae for the -Bernstein basis function, including the partition of unity property, and formulae for representing the monomials were studied. In addition, a kind of approximation of a function in terms of Bernoulli polynomials is used in several approaches for solving differential equations, such as [2][3][4]. This paper gives conditions to approximate capable functions as a linear combination of -Bernoulli polynomials as well as related examples.…”

Section: Introductionmentioning

confidence: 99%

Approximation Properties of q-Bernoulli Polynomials

Momenzadeh

Kakangi

2017

Abstract and Applied Analysis

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We study the -analogue of Euler-Maclaurin formula and by introducing a new -operator we drive to this form. Moreover, approximation properties of -Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of -Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.

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“…In [28], a new matrix method is proposed to solve two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions. In [29], the authors utilize the second kind Chebyshev wavelets to obtain the numerical solutions of the convection diffusion equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of One-Dimensional Fractional Nonsteady Heat Transfer Model Based on the Second Kind Chebyshev Wavelet

Zhao

Xie

Huang

2017

Discrete Dynamics in Nature and Society

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In the current study, a numerical technique for solving one-dimensional fractional nonsteady heat transfer model is presented. We construct the second kind Chebyshev wavelet and then derive the operational matrix of fractional-order integration. The operational matrix of fractional-order integration is utilized to reduce the original problem to a system of linear algebraic equations, and then the numerical solutions obtained by our method are compared with those obtained by CAS wavelet method. Lastly, illustrated examples are included to demonstrate the validity and applicability of the technique.

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A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions

Cited by 15 publications

References 36 publications

Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

Approximation Properties of q-Bernoulli Polynomials

Numerical Simulation of One-Dimensional Fractional Nonsteady Heat Transfer Model Based on the Second Kind Chebyshev Wavelet

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