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Numerical Scheme for the Computational Study of Two Dimensional Diffusion and Burgers' Systems with Stability and Error Estimate

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JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS 2025年第1期32卷 1-23页

作者： Bilal, Muhammad Ghafoor, Abdul Hussain, Manzoor Shah, Kamal Abdeljawad, Thabet Kohat Univ Sci & Technol Inst Numer Sci Kohat 26000 KP Pakistan Women Univ Azad Jammu & Kashmir Fac Sci & Technol Dept Math Computat & Appl Math Lab Bagh 12500 Pakistan Saveetha Univ Saveetha Inst Med & Tech Sci Saveetha Sch Engn Dept Math Chennai 602105 India Prince Sultan Univ Dept Math & Sci Riyadh 11586 Saudi Arabia China Med Univ Dept Med Res Taichung 40402 Taiwan Sefako Makgatho Hlth Sci Univ Sch Sci & Technol Dept Math & Appl Math Ga Rankuwa South Africa Univ Malakand Dept Math Chakdara KP Pakistan

This paper demonstrates a numerical stratagem for the solution of two dimensional single and coupled partial differential equations, using the new version of the Haar wavelets namely: the scale-3 Haar wavelets (S3HW), combined with the finite difference formulation. The proposed method consists of two phases. The first phase deals with the numerical estimation of the temporal derivative via finite difference which converts the problem to time discrete form. The second phase describes, the approximation of the spatial derivatives along with solution, adopting S3HW. Then, the collocation technique is implemented to transform the resultant system to the set of linear algebraic equations. Solution of the linear system gives the unknown wavelet coefficients which utilized to determine the numerical solutions. Afterwards, the error, convergence, and stability analysis are conducted and deduced a new error estimate. Besides, the numerical simulations are done to verify the scheme and the obtained theoretical findings (convergence and stability). To validate, the performance of the present scheme different error measures L-infinity, L-2, root mean square (RMS), and relative error (RE) are determined numerically. The scheme is also compared in terms of error with the scale-2 Haar wavelets and radial basis functions based algorithms. Overall judgement shows, that the numerical results of the developed scheme are in good agreement with the exact solution and the aforementioned methods in the literature.

关键词： Nonlinear PDEs Scale-3 Haar wavelet finite difference formulation Convergence and stability

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A novel approach to approximate solution of time fractional gas dynamics equation with Atangana-Baleanu derivative

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ALEXANDRIA ENGINEERING JOURNAL 2025年 116卷 451-471页

作者： Noureen, Rabia Iqbal, Muhammad Kashif Asgir, Maryam Almohsen, Bandar Azeem, Muhammad Govt Coll Univ Dept Math Faisalabad 38000 Pakistan Riphah Int Univ Dept Math Lahore Pakistan King Saud Univ Coll Sci Dept Math POB 2455 Riyadh 11451 Saudi Arabia Univ Manchester Sch Engn Dept Solids & Struct Manchester M13 9PL England

This research paper investigates a numerical scheme based on cubic B-spline functions for solving the time fractional gas dynamics equation incorporating the Atangana-Baleanu derivative. The Atangana-Baleanu time fractional derivative has been approximated by adopting the typical finite difference formulation while the cubic B-spline functions are employed to accomplish spatial discretization. A conditional bound for stability has been derived and a convergence analysis for cubic B-spline interpolation is conducted to measure the accuracy of the solution. The effectiveness and accuracy of the proposed method are tested through numerical simulations. Graphical and tabular results are exhibited to evaluate the outcomes of the proposed strategy. The major advantage of the suggested scheme is that the algorithm is straightforward to carry out.

关键词： Time fractional gas dynamics equation Atangana-Baleanu derivative Cubic B-spline functions Collocation method Stability finite difference formulation Convergence

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Experimental Investigation of the Effect of Wavefront Aberration on the Flow Generated by Laser-Induced Breakdown

Experimental Investigation of the Effect of Wavefront Aberra...

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AIAA SciTech Forum

作者： Noel, Zareb A. Gordeyev, Stanislav Rennie, R. Mark Univ Notre Dame Dept Aerosp & Mech Engn Hessert Lab Aerosp Res Notre Dame IN 46556 USA

ISBN: (数字)9781624107115

ISBN: (纸本)9781624107115

Phase aberrations and their effect on the development of flowfields caused by laser-induced breakdown is investigated. The phase aberrations are imparted onto a high-energy laser pulse at a wavelength of 1064nm using a deformable mirror. An experimental setup was devised to capture the motion of the flowfields resulting from laser-induced breakdown with phase aberration which focuses on capturing both the transverse profile, as well as the coaxial profile of the flowfield. It is shown that the laser pulse energy (181mJ) absorbed by the spark decreases significantly due to the presence of non-planar phase, which is due to spreading in the focal plane where breakdown typically occurs. Of the data collected, a single instance of Zernike ***, Zernike Y-Coma, and Zernike Spherical aberrations are investigated. The well-known Horn-Schunck optical flow method is used to analyze the shadowgraph images, producing dense optical flowfield representations of the motion. It is shown that each aberration investigated creates a unique flowfield, showing potential for hyper-specific local flow specification and the implications are further discussed.

关键词： Toroidal Vortex Laser Energy Velocity Profiles Flow Visualization Techniques Lagrange Multipliers Axisymmetric Flow Polarization Internal Combustion Engines Drag Reduction finite difference formulation

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An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana-Baleanu derivative

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ENGINEERING WITH COMPUTERS 2022年第1期38卷 901-917页

作者： Shafiq, Madiha Abbas, Muhammad Abualnaja, Khadijah M. Huntul, M. J. Majeed, Abdul Nazir, Tahir Univ Sargodha Dept Math Sargodha 40100 Pakistan Taif Univ Coll Sci Dept Math & Stat POB 11099 At Taif 21944 Saudi Arabia Jazan Univ Fac Sci Dept Math Jazan Saudi Arabia Univ Educ Div Sci & Technol Dept Math Lahore 54770 Pakistan

The present paper deals with cubic B-spline approximation together with theta- weighted scheme to obtain numerical solution of the time fractional advection diffusion equation using Atangana-Baleanu derivative. To discretize the Atangana-Baleanu time derivative containing a non-singular kernel, finite difference scheme is utilized. The cubic basis functions are associated with spatial discretization. The current discretization scheme used in the present study is unconditionally stable and the convergence is of order O(h(2) +Delta t(2)). The proposed scheme is validated through some numerical examples which reveal the current scheme is feasible and quite accurate.

关键词： Advection diffusion equation Cubic B-spline functions Spline interpolation Atangana-Baleanu time fractional derivative finite difference formulation Stability and convergence

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Steady-State Low-Order Explicit (LOE) Runge-Kutta Schemes with Improved Convergence

Steady-State Low-Order Explicit (LOE) Runge-Kutta Schemes wi...

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AIAA Aviation Forum

作者： Sabri, Zaid H. Hixon, Ray Univ Toledo Mech Ind & Mfg Engn MIME Dept Toledo OH 43606 USA

ISBN: (数字)9781624107047

ISBN: (纸本)9781624107047

Time marching techniques constitute a significant area of interest within Computational Aeroacoustics. One common time integration technique is the explicit Runge-Kutta scheme, which is used to obtain high orders of accuracy through a multistage procedure. Researchers have used and optimized many stages to obtain high accuracy and inviscid stability limits. Recent research has shown that high-staged Runge-Kutta schemes accuracy degrades near the boundaries. In this work, Low-Order Explicit (LOE) Runge-Kutta schemes are optimized to achieve high inviscid stability limits. Single-step and multi-step schemes are compared to higher-order schemes. The resulting schemes obtain nearly 90% of the stability limit per stage of the optimized Jameson five-stage scheme while using 50% fewer stages per time step and over 80% of the stability limit per stage of the Allampalli HALE-RK67 scheme while using 60% fewer stages per time step. Due to their low accuracy, the proposed schemes are designed to be part of an iterative residual method and will be validated in steady flow and implicit time marching applications.

关键词： Flow Conditions Computational Aeroacoustics Computational Fluid Dynamics Numerical Integration Euler Equations Wavenumber Acoustic Wave Convection NASA finite difference formulation

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New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

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ENGINEERING COMPUTATIONS 2021年第1期38卷 83-106页

作者： Nazir, Tahir Abbas, Muhammad Iqbal, Muhammad Kashif Univ Sargodha Dept Math Sargodha Pakistan Govt Coll Univ Dept Math Faisalabad Pakistan

Purpose The purpose of this paper is to present a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers' equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions and aerofoil flow theory. Design/methodology/approach The system of partial differential equations is discretized in time direction using the finite difference formulation, and the new CBS approximations have been used to interpolate the solution curves in the spatial direction. The theoretical estimation of stability and uniform convergence of the proposed numerical algorithm has been derived rigorously. Findings A different scheme based on the new approximation in CBS functions is proposed which is quite different from the existing methods developed (Mittal and Jiwari, 2012;Mittal and Arora, 2011;Mittal and Tripathi, 2014;Raslanet al., 2017;Shallalet al., 2019). Some numerical examples are presented to validate the performance and accuracy of the proposed technique. The simulation results have guaranteed the superior performance of the presented algorithm over the existing numerical techniques on approximate solutions of coupled viscous Burgers' equations. Originality/value The current approach based on new CBS approximations is novel for the numerical study of coupled Burgers' equations, and as far as we are aware, it has never been used for this purpose before.

关键词： Coupled viscous Burgers' equations Cubic B-spline functions finite difference formulation Crank-Nicolson scheme Von-Neumann stability analysis

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MEMORY EFFECT ANALYSIS USING PIECEWISE CUBIC B-SPLINE OF TIME FRACTIONAL DIFFUSION EQUATION

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FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2022年第8期30卷 2240270-2240270页

作者： Shafiq, Madiha Abdullah, Farah Aini Abbas, Muhammad Alzaidi, Ahmed Sm Riaz, Muhammad Bilal Univ Sargodha Dept Math Sargodha Pakistan Sch Math Sci Univ Sains Malaysia George Town 11800 Malaysia Taif Univ Coll Sci Dept Math & Stat POB 11099 Taif 21944 Saudi Arabia Gdansk Univ Technol Inst Appl Math Fac Appl Phys & Math PL-80233 Gdansk Poland

The purpose of this work is to study the memory effect analysis of Caputo-Fabrizio time fractional diffusion equation by means of cubic B-spline functions. The Caputo-Fabrizio interpretation of fractional derivative involves a non-singular kernel that permits to describe some class of material heterogeneities and the effect of memory more effectively. The proposed numerical technique relies on finite difference approach and cubic B-spline functions for discretization along temporal and spatial grids, respectively. To ensure that the error does not amplify during computational process, stability analysis is performed. The described algorithm is second-order convergent along time and space directions. The computational competence of the scheme is tested through some numerical examples. The results reveal that the current scheme is reasonably efficient and reliable to be used for solving the subject problem.

关键词： Diffusion Equation Spline Interpolation Caputo-Fabrizio Fractional Derivative Cubic B-Spline Functions Stability finite difference formulation Convergence

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A computational approach for solving time fractional differential equation via spline functions

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ALEXANDRIA ENGINEERING JOURNAL 2020年第5期59卷 3061-3078页

作者： Khalid, Nauman Abbas, Muhammad Iqbal, Muhammad Kashif Singh, Jagdev Ismail, Ahmad Izani Md. Natl Coll Business Adm & Econ Dept Math Lahore Pakistan Univ Sargodha Dept Math Sargodha Pakistan Govt Coll Univ Dept Math Faisalabad Pakistan JECRC Univ Dept Math Jaipur Rajasthan India Univ Sains Malaysia Sch Math Sci George Town Malaysia

A computational approach based on finite difference scheme and a redefined extended B-spline functions is presented to study the approximate solution of time fractional advection diffusion equation. The Caputo time-fractional derivative and redefined extended B-spline functions have been used for the time and spatial discretization, respectively. The numerical scheme is shown to be O(h(2) + Delta t(2-alpha)) accurate and unconditionally stable. The proposed method is tested through some numerical experiments involving homogeneous/non-homogeneous boundary conditions which concluded that it is more accurate than existing methods. The simulation results show superior agreement with the exact solution as compared to existing methods. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.

关键词： Redefined extended cubic B-spline functions Caputo's time fractional derivative Advection-diffusion equation Stability and convergence finite difference formulation

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Solving time fractional Burgers' and Fisher's equations using cubic B-spline approximation method

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ADVANCES IN difference EQUATIONS 2020年第1期2020卷 1-15页

作者： Majeed, Abdul Kamran, Mohsin Iqbal, Muhammad Kashif Baleanu, Dumitru Univ Educ Dept Math Div Sci & Technol Lahore Pakistan Govt Coll Univ Dept Math Faisalabad Pakistan Cankaya Univ Fac Arts & Sci Dept Math Ankara Turkey China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan Inst Space Sci Magurele Romania

This article presents a numerical algorithm for solving time fractional Burgers' and Fisher's equations using cubic B-spline finite element method. The L1 formula with Caputo derivative is used to discretized the time fractional derivative, whereas the Crank-Nicolson scheme based on cubic B-spline functions is used to interpolate the solution curve along the spatial grid. The numerical scheme has been implemented on three test problems. The obtained results indicate that the proposed method is a good option for solving nonlinear fractional Burgers' and Fisher's equations. The error norms L2 and L infinity have been calculated to validate the efficiency and accuracy of the presented algorithm.

关键词： Cubic B-spline collocation method Time fractional differential equation Caputo's fractional derivative Stability and convergence finite difference formulation

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A numerical investigation of Caputo time fractional Allen-Cahn equation using redefined cubic B-spline functions

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ADVANCES IN difference EQUATIONS 2020年第1期2020卷 1-22页

作者： Khalid, Nauman Abbas, Muhammad Iqbal, Muhammad Kashif Baleanu, Dumitru Natl Coll Business Adm & Econ Dept Math Lahore Pakistan Ton Duc Thang Univ Informetr Res Grp Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Math & Stat Ho Chi Minh City Vietnam Univ Sargodha Dept Math Sargodha Pakistan Govt Coll Univ Dept Math Faisalabad Pakistan Cankaya Univ Fac Arts & Sci Dept Math Ankara Turkey China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan Inst Space Sci Magurele Romania

We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate solution of time fractional Allen-Cahn equation (ACE). We discretize the time fractional derivative of order alpha is an element of (0,1] by using finite forward difference formula and bring RCBS functions into action for spatial discretization. We find that the numerical scheme is of order O(h2+Delta t2-alpha) and unconditionally stable. We test the computational efficiency of the proposed method through some numerical examples subject to homogeneous/nonhomogeneous boundary constraints. The simulation results show a superior agreement with the exact solution as compared to those found in the literature.

关键词： Redefined cubic B-spline functions Time fractional Allen-Cahn equation Caputo's time fractional derivative Stability and convergence finite difference formulation

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