A numerical approach for solving weakly singular partial integro‐differential equations via two‐dimensional‐orthonormal Bernstein polynomials with the convergence analysis

Mirzaee, Farshid; Alipour, Sahar; Samadyar, Nasrin

doi:10.1002/num.22316

Cited by 39 publications

(25 citation statements)

References 32 publications

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“…Definition The representation of the 1D‐OBPs of degree n on the interval [0,1] in terms of original nonorthonormal Bernstein basis functions is as follows:

ϕ_{i, n} false(s false) = \sqrt{2 false(n - i false) + 1} true \sum_{k = 0}^{i} {false(- 1 false)}^{k} \frac{((), \binom{2 n + 1 - k}{i - k}) ((), \binom{i}{k})}{((), \binom{n - k}{i - k})} B_{i - k, n - k} false(s false), i = 0, 1, ⋯, n,

where

B_{i, n} false(s false) = true \sum_{k = 0}^{n - i} {false(- 1 false)}^{k} ((), \binom{n}{i}) ((), \binom{n - i}{k}) s^{i + k}, i = 0, 1, ⋯, n .

Let Φ( s ) be a set of 1D‐OBPs of degree n as follows:

normalΦ false(s false) = {false[ϕ_{0, n} false(s false), ϕ_{1, n} false(s false), ⋯, ϕ_{n, n} false(s false) false]}^{T},

where ϕ r , n ( s ) for r = 0,1,…, n , defined in Equation . The vector Φ( s ) is displayed based on standard basis in the following matrix form:

normalΦ false(s false) = A T_{n} false(s false), x \in false[0, 1 false],

where

…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…Finally, we obtain the 2D‐FOBPs operational matrices of integration, product, and fractional partial integration and derivative. Also, I and O be the identity and zero matrices of order ( n + 1), respectively, and ⊗ denote the Kronecker product …”

Section: Operational Matricesmentioning

confidence: 99%

“…Definition 7. The representation of the 1D-OBPs of degree n on the interval [0, 1] in terms of original nonorthonormal Bernstein basis functions is as follows 24,32 :…”

Section: Orthonormal Bernstein Polynomialsmentioning

confidence: 99%

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Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Mirzaee

Alipour

2019

Math Methods in App Sciences

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In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.

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“…Definition The representation of the 1D‐OBPs of degree n on the interval [0,1] in terms of original nonorthonormal Bernstein basis functions is as follows:

ϕ_{i, n} false(s false) = \sqrt{2 false(n - i false) + 1} true \sum_{k = 0}^{i} {false(- 1 false)}^{k} \frac{((), \binom{2 n + 1 - k}{i - k}) ((), \binom{i}{k})}{((), \binom{n - k}{i - k})} B_{i - k, n - k} false(s false), i = 0, 1, ⋯, n,

where

B_{i, n} false(s false) = true \sum_{k = 0}^{n - i} {false(- 1 false)}^{k} ((), \binom{n}{i}) ((), \binom{n - i}{k}) s^{i + k}, i = 0, 1, ⋯, n .

Let Φ( s ) be a set of 1D‐OBPs of degree n as follows:

normalΦ false(s false) = {false[ϕ_{0, n} false(s false), ϕ_{1, n} false(s false), ⋯, ϕ_{n, n} false(s false) false]}^{T},

where ϕ r , n ( s ) for r = 0,1,…, n , defined in Equation . The vector Φ( s ) is displayed based on standard basis in the following matrix form:

normalΦ false(s false) = A T_{n} false(s false), x \in false[0, 1 false],

where

…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Section: Operational Matricesmentioning

confidence: 99%

See 1 more Smart Citation

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Mirzaee

Alipour

2019

Math Methods in App Sciences

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“…Also, there are few numerical methods for solving partial integro‐differential equations. For instance, an efficient matrix method based on two‐dimensional orthonormal Bernstein polynomials have been presented to solve linear and nonlinear weakly singular partial integro‐differential equations in Mirzaee et al Numerical solution of a Volterra integro‐differential equation of parabolic type with memory term subject to initial boundary value conditions has been presented in Fahim et al Spline collocation methods have been proposed for the spatial discretization of a class of hyperbolic partial integro‐differential equations in Fairweather …”

Section: Introductionmentioning

confidence: 99%

“…9 Collocation method based on the Bernstein polynomials has been introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel in Işik et al 10 Also, there are few numerical methods for solving partial integro-differential equations. For instance, an efficient matrix method based on two-dimensional orthonormal Bernstein polynomials have been presented to solve linear and nonlinear weakly singular partial integro-differential equations in Mirzaee et al 11 Numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions has been presented in Fahim et al 12 Spline collocation methods have been proposed for the spatial discretization of a class of hyperbolic partial integro-differential equations in Fairweather. 13 In recent years, it has become obvious that real-life problems, such as fluid mechanics, biology, physics, and physiology, [14][15][16] can be more satisfactorily modeled by different types of fractional equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Samadyar

Mirzaee

2019

Int J Numerical Modelling

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In this paper, an efficient matrix method based on 2D orthonormal Bernoulli polynomials are developed to obtain numerical solution of weakly singular fractional partial integro‐differential equations (FPIDEs). Operational matrix of integration, almost operational matrix of integration, and product operational matrix are employed to transform the solution of Volterra singular FPIDEs to the system of linear or nonlinear algebraic equations. The obtained algebraic system can be easily solved by using an appropriate numerical method. Also, some useful theorems concerning the convergence and error analysis associated to the mentioned approach are proven in the current paper. Finally, some numerical examples are included to illustrate the accuracy, efficiency, and applicability of the proposed approach.

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Piecewise barycentric interpolating functions for the numerical solution of Volterra integro‐differential equations

Torkaman

Heydari

Loghmani

2022

Math Methods in App Sciences

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This investigation presents an effective numerical scheme using a new set of basis functions, namely, the piecewise barycentric interpolating functions, to find the approximate solution of Volterra integro-differential equations (VIDEs).The operational matrices of integration and product for the PBIFs are provided. Then these operational matrices are utilized to reduce the VIDEs to a system of algebraic equations. Applying the Floater-Hormann weights, the convergence analysis of the PBIFs method is studied. Finally, several numerical examples are provided to illustrate the efficiency and validity of the proposed method in acceptable computational times, and the results are compared with some existing numerical methods. KEYWORDS convergence analysis, Floater-Hormann interpolant, operational matrices of integral and product, piecewise barycentric interpolating functions, Volterra integro-differential equations MSC CLASSIFICATION 45D05; 45G10; 65R20; 65D05

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A numerical approach for solving weakly singular partial integro‐differential equations via two‐dimensional‐orthonormal Bernstein polynomials with the convergence analysis

Abstract: KEYWORDS error analysis, operational matrix method, singular integral equation, two-dimensional orthonormal Bernstein polynomials Numer Methods Partial Differential Eq. 2019;35:615-637. wileyonlinelibrary.com/journal/num

Cited by 39 publications

References 32 publications

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Piecewise barycentric interpolating functions for the numerical solution of Volterra integro‐differential equations

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