In this article,we develop the Laplace transform(LT)based Chebyshev spectral collocation method(CSCM)to approximate the time fractional advection-diffusion equation,incorporating the Atangana-Baleanu Caputo(ABC)deriva...In this article,we develop the Laplace transform(LT)based Chebyshev spectral collocation method(CSCM)to approximate the time fractional advection-diffusion equation,incorporating the Atangana-Baleanu Caputo(ABC)derivative.The advection-diffusion equation,which governs the transport of mass,heat,or energy through combined advection and diffusion processes,is central to modeling physical systems with nonlocal behavior.Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain,eliminating the need for classical time-stepping methods that often suffer from stability constraints.For spatial discretization,we employ the CSCM,where the solution is approximated using Lagrange interpolation polynomial based on the Chebyshev collocation nodes,achieving exponential convergence that outperforms the algebraic convergence rates of finite difference and finite element methods.Finally,the solution is reverted to the time domain using contour integration technique.We also establish the existence and uniqueness of the solution for the proposed problem.The performance,efficiency,and accuracy of the proposed method are validated through various fractional advection-diffusion problems.The computed results demonstrate that the proposed method has less computational cost and is highly accurate.展开更多
This study presents a numerical investigation of shallow water wave dynamics with particular emphasis on the role of surface tension.In the absence of surface tension,shallow water waves are primarily driven by gravit...This study presents a numerical investigation of shallow water wave dynamics with particular emphasis on the role of surface tension.In the absence of surface tension,shallow water waves are primarily driven by gravity and are well described by the classical Boussinesq equation,which incorporates fourth-order dispersion.Under this framework,solitary and shock waves arise through the balance of nonlinearity and gravity-induced dispersion,producing waveforms whose propagation speed,amplitude,and width depend largely on depth and initial disturbance.The resulting dynamics are comparatively smoother,with solitary waves maintaining coherent structures and shock waves displaying gradual transitions.When surface tension is incorporated,however,the dynamics become significantly richer.Surface tension introduces additional sixth-order dispersive terms into the governing equation,extending the classical model to the sixth-order Boussinesq equation.This higher-order dispersion modifies the balance between nonlinearity and dispersion,leading to sharper solitary wave profiles,altered shock structures,and a stronger sensitivity of wave stability to parametric variations.Surface tension effects also change the scaling laws for wave amplitude and velocity,producing conditions where solitary waves can narrow while maintaining large amplitudes,or where shock fronts steepen more rapidly compared to the tension-free case.These differences highlight how capillary forces,though often neglected in macroscopic wave studies,play a fundamental role in shaping dynamics at smaller scales or in systems with strong fluid–interface interactions.The analysis in this work is carried out using the Laplace-Adomian Decomposition Method(LADM),chosen for its efficiency and accuracy in solving high-order nonlinear partial differential equations.The numerical scheme successfully recovers both solitary and shock wave solutions under the sixth-order model,with error analysis confirming remarkably low numerical deviations.These results underscore the robustness of the method while demonstrating the profound contrast between shallow water wave dynamics without and with surface tension.展开更多
针对Heisenberg群上p-次Laplace方程,建立其弱解的Liouville型定理,证明过程主要基于Moser迭代技巧和弱解的正则性结果。In this paper, we give a Liouville type theorem for the weak solution of the p-sub-Laplacian equation on t...针对Heisenberg群上p-次Laplace方程,建立其弱解的Liouville型定理,证明过程主要基于Moser迭代技巧和弱解的正则性结果。In this paper, we give a Liouville type theorem for the weak solution of the p-sub-Laplacian equation on the Heisenberg group. The proof process relies on Moser iterative techniques and the regularity results of weak solutions.展开更多
基金extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/174/46.
文摘In this article,we develop the Laplace transform(LT)based Chebyshev spectral collocation method(CSCM)to approximate the time fractional advection-diffusion equation,incorporating the Atangana-Baleanu Caputo(ABC)derivative.The advection-diffusion equation,which governs the transport of mass,heat,or energy through combined advection and diffusion processes,is central to modeling physical systems with nonlocal behavior.Our numerical scheme employs the LT to transform the time-dependent time-fractional PDEs into a time-independent PDE in LT domain,eliminating the need for classical time-stepping methods that often suffer from stability constraints.For spatial discretization,we employ the CSCM,where the solution is approximated using Lagrange interpolation polynomial based on the Chebyshev collocation nodes,achieving exponential convergence that outperforms the algebraic convergence rates of finite difference and finite element methods.Finally,the solution is reverted to the time domain using contour integration technique.We also establish the existence and uniqueness of the solution for the proposed problem.The performance,efficiency,and accuracy of the proposed method are validated through various fractional advection-diffusion problems.The computed results demonstrate that the proposed method has less computational cost and is highly accurate.
文摘This study presents a numerical investigation of shallow water wave dynamics with particular emphasis on the role of surface tension.In the absence of surface tension,shallow water waves are primarily driven by gravity and are well described by the classical Boussinesq equation,which incorporates fourth-order dispersion.Under this framework,solitary and shock waves arise through the balance of nonlinearity and gravity-induced dispersion,producing waveforms whose propagation speed,amplitude,and width depend largely on depth and initial disturbance.The resulting dynamics are comparatively smoother,with solitary waves maintaining coherent structures and shock waves displaying gradual transitions.When surface tension is incorporated,however,the dynamics become significantly richer.Surface tension introduces additional sixth-order dispersive terms into the governing equation,extending the classical model to the sixth-order Boussinesq equation.This higher-order dispersion modifies the balance between nonlinearity and dispersion,leading to sharper solitary wave profiles,altered shock structures,and a stronger sensitivity of wave stability to parametric variations.Surface tension effects also change the scaling laws for wave amplitude and velocity,producing conditions where solitary waves can narrow while maintaining large amplitudes,or where shock fronts steepen more rapidly compared to the tension-free case.These differences highlight how capillary forces,though often neglected in macroscopic wave studies,play a fundamental role in shaping dynamics at smaller scales or in systems with strong fluid–interface interactions.The analysis in this work is carried out using the Laplace-Adomian Decomposition Method(LADM),chosen for its efficiency and accuracy in solving high-order nonlinear partial differential equations.The numerical scheme successfully recovers both solitary and shock wave solutions under the sixth-order model,with error analysis confirming remarkably low numerical deviations.These results underscore the robustness of the method while demonstrating the profound contrast between shallow water wave dynamics without and with surface tension.
文摘针对Heisenberg群上p-次Laplace方程,建立其弱解的Liouville型定理,证明过程主要基于Moser迭代技巧和弱解的正则性结果。In this paper, we give a Liouville type theorem for the weak solution of the p-sub-Laplacian equation on the Heisenberg group. The proof process relies on Moser iterative techniques and the regularity results of weak solutions.
