Initial‐boundary value problem with a nonlocal condition for             a viscosity equation

Bouziani, Abdelfatah

doi:10.1155/s0161171202004167

Cited by 22 publications

(21 citation statements)

References 8 publications

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“…According to [27], the space W (3,1) is a reproducing kernel Hilbert space and similar to [28], we can obtain its reproducing kernel R (3,1) ( …”

Section: G the Space W (31) (D)mentioning

confidence: 99%

“…Remark 2.2. We remark that, for every u(x, t) ∈ W (4,2) u( W (3,1) Now, we will give the representation of analytical solution to (6)- (9) in W (3,1) .…”

Section: H the Space W (42) (D)mentioning

confidence: 99%

“…Now, suppose that ϑ i (x, t) ∞ i=1 be a normalized orthogonal system in W (3,1) , such that ϑ i (x, t) = i k=1 δ ik ϑ k (x, t), i = 1, 2, · · · , then we state the following theorem that gives the exact solution. Theorem 3.3 (see [26]).…”

Section: Lemma 32 (See [29]) Letmentioning

confidence: 99%

“…We now study the convergence of proposed method under some hypotheses which provide the theoretical basis of the proposed method and further present the error estimates for obtained approximation in W (3,1) .…”

Section: A Convergence and Error Analysis In W (31)mentioning

confidence: 99%

“…The study of boundary value problems with integral conditions has attracted many researchers due to its wide application in engineering in many areas, see, for example, [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

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Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

Fardi

Ghasemi

2016

Numerical Methods Partial

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This article is concerned with a method for solving nonlocal initial-boundary value problems for parabolic and hyperbolic integro-differential equations in reproducing kernel Hilbert space. Convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method and some error estimates for the numerical approximation in reproducing kernel Hilbert space are presented. Finally, two numerical examples are considered to illustrate the computation efficiency and accuracy of the proposed method.

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“…According to [27], the space W (3,1) is a reproducing kernel Hilbert space and similar to [28], we can obtain its reproducing kernel R (3,1) ( …”

Section: G the Space W (31) (D)mentioning

confidence: 99%

“…Remark 2.2. We remark that, for every u(x, t) ∈ W (4,2) u( W (3,1) Now, we will give the representation of analytical solution to (6)- (9) in W (3,1) .…”

Section: H the Space W (42) (D)mentioning

confidence: 99%

Section: Lemma 32 (See [29]) Letmentioning

confidence: 99%

Section: A Convergence and Error Analysis In W (31)mentioning

confidence: 99%

“…The study of boundary value problems with integral conditions has attracted many researchers due to its wide application in engineering in many areas, see, for example, [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

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Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

Fardi

Ghasemi

2016

Numerical Methods Partial

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Reduction of Cr(VI) to Cr(III) and removal of total chromium from wastewater using scrap iron in the form of zerovalent iron(ZVI): Batch and column studies

2011

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This work experimentally investigates Cr(VI) reduction to Cr(III) using waste scrap iron in the form of zerovalent iron (ZVI) collected from the mechanical workshop of the Institute, both in batch and continuous operation. The reduction of Cr(VI) to Cr(III) was found to be complete (∼100%) depending on the experimental conditions. Lower pH values favour Cr(VI) reduction. Two concurrent reactions take place, that is reduction of Cr(VI) by Fe 0 (ZVI) and by Fe 2+ generated due to H + corrosion of iron. Maximum around 22%, 11% and 2% Cr(III) remained dissolved in solution while the experiments were carried out at initial pH of 2, 4.67 and 7. Higher ZVI loading increases Cr(VI) reduction rate, however, consumption of iron is noted to be higher. The results indicate that the bed is exhausted rapidly at higher pH, initial Cr(VI) concentration and flow rate. This is attributable to predominance passivation of ZVI surface forming Cr(III)-Fe(III)-oxide layer. SEM analysis of ZVI before and after the experiments confirms formation of passive oxide on iron surface is responsible for deterioration of Cr(VI) reduction efficiency due to its blanketing effect.

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On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Dehghan

2004

Numerical Methods Partial

199

123

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Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article.

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Initial‐boundary value problem with a nonlocal condition for a viscosity equation

Cited by 22 publications

References 8 publications

Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

Reduction of Cr(VI) to Cr(III) and removal of total chromium from wastewater using scrap iron in the form of zerovalent iron(ZVI): Batch and column studies

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

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