Numerical Algorithm Based on Haar-Sinc Collocation Method for Solving the Hyperbolic PDEs

Pirkhedri, A.; Javadi, Hamid Haj Seyyed; Navidi, Hamidreza

doi:10.1155/2014/340752

Cited by 5 publications

(8 citation statements)

References 32 publications

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“…Since we will replace the first derivative by it matrix approximations, here we define the m × m matrix I (1) whose in − th entry is given by δ (1) i−n . An important class of functions denoted by L α (D) play crucial rule in approximating derivatives will be needed in next formula (see, [17]…”

Section: Sinc-approximation Formulamentioning

confidence: 99%

“…that turn (1) and (3) into a system that is of mixed type (hyperbolic-elliptic). In recent years, the numerical (approximate) solutions for (1) was considered by many authors.…”

Section: Introductionmentioning

confidence: 99%

“…that turn (1) and (3) into a system that is of mixed type (hyperbolic-elliptic). In recent years, the numerical (approximate) solutions for (1) was considered by many authors. In [1] they investigates the Haar-Sinc collocation method for the solution of hyperbolic systems of partial differential equations with applications to the telegraph equation.…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, the numerical (approximate) solutions for (1) was considered by many authors. In [1] they investigates the Haar-Sinc collocation method for the solution of hyperbolic systems of partial differential equations with applications to the telegraph equation. Existence and uniqueness for a 2 × 2 system of conservation laws, called Brio system, that arising as a model used in magnetohydrodynamics is developed in [2].…”

Section: Introductionmentioning

confidence: 99%

“…A new convergence proof for the solution conservation laws of mixed type using the ADM was proposed in [11]. This is with the knowledge that there are many previous studies dealing with this topic, and some of them gave examples from physics and other sciences, to demonstrate the importance of system appeared in (1), and how it is used in practical life, for example we refer to [3,[12][13][14].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions

Al‐Khaled¹,

Rababah²

2020

Symmetry

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In this paper, we develop an accurate technique via the use of the Adomian decomposition method (ADM) to solve analytically a 2 × 2 systems of partial differential equation that represent balance laws of hyperbolic-elliptic type. We prove that the sequence of iteration obtained by ADM converges strongly to the exact solution by establishing a construction of fixed points. For comparison purposes, we also use the Sinc function methodology to establish a new procedure to solve numerically the same system. It is shown that approximation by Sinc function converges to the exact solution exponentially, also handles changes in type. A numerical example is presented to demonstrate the theoretical results. It is noted that the two methods show the symmetry in the approximate solution. The results obtained by both methods reveal that they are reliable and convenient for solving balance laws where the initial conditions are of the Riemann type.

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Section: Sinc-approximation Formulamentioning

confidence: 99%

“…that turn (1) and (3) into a system that is of mixed type (hyperbolic-elliptic). In recent years, the numerical (approximate) solutions for (1) was considered by many authors.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions

Al‐Khaled¹,

Rababah²

2020

Symmetry

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Numerical solution of the one dimensional Schrödinger equation using a basis set of scaled and shifted sinc functions on a finite interval

Braun

2023

Journal of Computational and Applied Mathematics

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A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations

et al. 2022

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Usually, to find the analytical and numerical solution of the boundary value problems of fractional partial differential equations is not an easy task; however, the researchers devoted their sincere attempt to find the solutions of various equations by using either analytical or numerical procedures. In this article, a very accurate and prominent method is developed to find the analytical solution of hyperbolic-telegraph equations with initial and boundary conditions within the Caputo operator, which has very simple calculations. This method is called a new technique of Adomian decomposition method. The obtained results are described by plots to confirm the accuracy of the suggested technique. Plots are drawn for both fractional and integer order solutions to confirm the accuracy and validity of the proposed method. Solutions are obtained at different fractional orders to discuss the useful dynamics of the targeted problems. Moreover, the suggested technique has provided the highest accuracy with a small number of calculations. The suggested technique gives results in the form of a series of solutions with easily computable and convergent components. The method is simple and straightforward and therefore preferred for the solutions of other problems with both initial and boundary conditions.

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Numerical Algorithm Based on Haar-Sinc Collocation Method for Solving the Hyperbolic PDEs

Cited by 5 publications

References 32 publications

Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions

Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions

Numerical solution of the one dimensional Schrödinger equation using a basis set of scaled and shifted sinc functions on a finite interval

A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations

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