A Numerical Solution of the Advection-Diffusion Equation by Using Extended Cubic B-Spline Functions

Görgülü, Melis Zorşahin; Dağ, İdris; Dogan, Sumeyye; Irk, Dursun

doi:10.18038/aubtda.336116

Cited by 4 publications

(6 citation statements)

References 20 publications

(7 reference statements)

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“…In this section, the motion of the single solitary wave and the interaction of two solitary waves are studied to validate the efficiency and applicability of the suggested algorithm. Accuracy of solution is checked by evaluating error norm (11) and the temporal order of convergence is worked out by the formula order (12) where represents the error norm for temporal step…”

Section: Resultsmentioning

confidence: 99%

“…From Figure , it is observed that the solitary wave propagates to the right keeping its original shape. A comparison of the results obtained by the proposed algorithm with the some existing techniques given in [2,3,7,11,12,13,14,15] is provided in Table . Comparison verifies that the suggested algorithm gives much better results than the other techniques given in Table . The conservation invariants, the temporal rate of convergence and the error norm are given in Table 2. It can be noticed from Table that for the fixed space step, when the temporal step size is decreased from to , the temporal order of convergence is almost three and the calculated invariants are in almost good agreement with their theoretical values.…”

Section: The Motion Of the Single Solitary Wavementioning

confidence: 99%

“…It can be noticed from Table that for the fixed space step, when the temporal step size is decreased from to , the temporal order of convergence is almost three and the calculated invariants are in almost good agreement with their theoretical values. The plot of absolute error with , is exhibited in 4.84 [3] 1.65 [7] 0.80 [11] 1.08 [12] 0.85 [13] 9.06 [14] 5.44 [15] 1.25 Exact…”

Section: The Motion Of the Single Solitary Wavementioning

confidence: 99%

“…Therefore, several computational techniques have been proposed to get the approximate solutions of the MRLW equation. Some of those methods are finite difference technique [1,2,3], the homotopy perturbation approach [4], the second-order Fourier pseudospectral method [5], the new approach based on the homotopy analysis [6], the moving least square collocation approach [7], the meshless method [8], Petrov-Galerkin method [9], subdomain technique with quartic B-spline functions [10], Galerkin method based on various B-spline functions [11,12], collocation approach with various Bsplines [13,14,15,16,17,18]. The main objective of the present work is to develop a numerical algorithm to obtain highly accurate numerical solutions of the MRLW equation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Modified Regularized Long Wave Denkleminin Nümerik Çözümü İçin Kuintik Trigonometrik B-spline Algoritması

Kirli

2023

Eskişehir Türk Dünyası Uygulama Ve Araştırma Merkezi Bilişim Dergisi

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This study introduces a new numerical algorithm for solving the modified regularized long-wave (MRLW) equation. To discretize the spatial variables and their derivatives, the collocation technique with quintic trigonometric B-spline functions is utilized and for the temporal derivative, the Adam's Moulton scheme is implemented. The performance and efficiency of the computational algorithm is tested on sample problem including the motion of single solitary wave. The error norm and three conservation constants are computed and compared with some of those available in the literature. The computed results verify that the suggested algorithm has the advantage in obtaining a highly accurate approximate solution of the MRLW equation as compared to the existing methods.

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Section: Resultsmentioning

confidence: 99%

Section: The Motion Of the Single Solitary Wavementioning

confidence: 99%

Section: The Motion Of the Single Solitary Wavementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Modified Regularized Long Wave Denkleminin Nümerik Çözümü İçin Kuintik Trigonometrik B-spline Algoritması

Kirli

2023

Eskişehir Türk Dünyası Uygulama Ve Araştırma Merkezi Bilişim Dergisi

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“…From the literature, it has been learned that scientists have presented numerous discretization techniques to obtain stable, efficient numerical schemes. In [7] Gorgulu et al extended cubic B-spline Galerkin arrangement in combination with the second-order Crank-Nicolson central difference scheme is used to solve the one-dimensional linear advection-diffusion (1D-LAD) equation; however, it experiences oscillations. In recent years, B-splines have gotten a lot of attention from researchers because they produce accurate results [8,9]; these splines are complicated but require less computation time and effort.…”

Section: Introductionmentioning

confidence: 99%

A stable r-adaptive mesh technique to analyze the advection-diffusion equation

2023

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This paper offers a study of the moving mesh method employed in one-dimensional linear and nonlinear advection-diﬀusion equations with different boundary and initial conditions. Advection and diffusion appear in the crux of the physical processes, where the transport of heat or other physical variables evolves. The aim is to present an accurate, stable moving finite-difference meshing scheme with its convergence. The velocity-profile of the considered cases is non-linear; therefore, the difference scheme needs mesh refinement. The moving mesh method analyzes the problem physics and adjusts the mesh according to the problem as it moves nodes in the region of more fluctuations. The approximate numerical results are estimated using four moving mesh partial differential equations with varying numbers of nodes. The moving mesh method is an r-adaptive technique that uses a fixed number of mesh nodes and moves the grids where error reduction is needed. The numerical solutions obtained are compared with the analytical solutions cited from the literature. The study presents five cases dealing with linear and non-linear examples in detail to understand the physics of the problem. The presented difference scheme is considerably more efficient than the numerical methods given in the literature.

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Various optimized artificial neural network simulations of advection-diffusion processes

Sari,

Gulen,

Celenk

2024

Phys. Scr.

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The aim of this research is to describe an artificial neural network (ANN) based method to approximate the solutions of the natural advection-diffusion equations. Although the solutions of these equations can be obtained by various effective numerical methods, feed forward neural network (FFNN) techniques combined with different optimization techniques offer a more practicable and flexible alternative than the traditional approaches to solve those equations. However, the ability of FFNN techniques to solve partial differential equations is a questionable issue and has not yet been fully concluded in the existing literature. The reliability and accuracy of computational results can be advanced by the choice of optimization techniques. Therefore, this study aims to take an effective step towards presenting the ability to solve the advection-diffusion equations by leveraging the inherent benefits of ANN methods while avoiding some of the limitations of traditional approaches. In this technique, the solution process requires minimizing the error generated by using a differential equation whose solution is considered as a trial solution. More specifically, this study uses a FFNN and backpropagation technique, one of the variants of the ANN method, to minimize the error and the adjustment of parameters. In the solution process, the loss function (error) needs to be minimized; this is accomplished by fitting the trial function into the differential equation using appropriate optimization techniques and obtaining the network output. Therefore, in this study, the commonly used techniques in the literature, namely gradient descent (GD), particle swarm optimization (PSO) and artificial bee colony (ABC), are selected to compare the effectiveness of gradient and gradient-free optimization techniques in solving the advection-diffusion equation. The calculations with all three optimization techniques for linear and nonlinear advection-diffusion equations have been run several times to obtain the optimum accuracy of the results. The computed results are seen to be very promising and in good agreement with the effective numerical methods and the physics-informed neural network (PINN) method in the literature. It is also concluded that the PSO-based algorithm outperforms other methods in terms of accuracy.

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A Numerical Solution of the Advection-Diffusion Equation by Using Extended Cubic B-Spline Functions

Cited by 4 publications

References 20 publications

Modified Regularized Long Wave Denkleminin Nümerik Çözümü İçin Kuintik Trigonometrik B-spline Algoritması

Modified Regularized Long Wave Denkleminin Nümerik Çözümü İçin Kuintik Trigonometrik B-spline Algoritması

A stable r-adaptive mesh technique to analyze the advection-diffusion equation

Various optimized artificial neural network simulations of advection-diffusion processes

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