This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of...This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).展开更多
This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by...This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.展开更多
This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight ...This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation polynomials based on the zeros of polynomials with the leading coefficient 1 of the least deviation from zero in L_(p,ω)[-1,1]are optimal for 1≤p<∞.We also give the optimal Lagrange interpolation nodes when the endpoints are included in the nodes.展开更多
This paper deals with the description and the representation of polynomials defined over n-simplices, The polynomials are computed by using two recurrent schemes: the Neville-Aitken one for the Lagrange interpolating ...This paper deals with the description and the representation of polynomials defined over n-simplices, The polynomials are computed by using two recurrent schemes: the Neville-Aitken one for the Lagrange interpolating operator and the De Casteljau one for the Bernstein-Bezier approximating operator. Both schemes fall intothe framework of transformations of the form where the F iare given numbers (forexample, at the initial step they coincide with the values of the function on a given lattice), and the coefficients (x) are linear polynomials valued in x and x is fixed. A general theory for such sequence of transformations can be found in [2] where it is also proved that these tranformations are completely characterized in term of a linear functional, reference functional. This functional is associated with a linear space., characteristic space.The concepts of reference functionals and characteristic spaces will be used and we shall prove the existence of a characteristic space for the reference functional: associated with these operators.展开更多
为提高网络入侵检测系统中检测算法的分类精度,降低训练样本及学习时间,在基于支持向量回归机的基础上,提出一种新的利用Lagrange支持向量回归机设计IDS的检测算法。使用KDD CUP 1999数据集进行仿真实验,结果表明该算法较基于支持向量...为提高网络入侵检测系统中检测算法的分类精度,降低训练样本及学习时间,在基于支持向量回归机的基础上,提出一种新的利用Lagrange支持向量回归机设计IDS的检测算法。使用KDD CUP 1999数据集进行仿真实验,结果表明该算法较基于支持向量回归机的检测算法具有更良好的泛化性能、更快的迭代速度、更高的检测精度和更低的误报率。展开更多
Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynom...Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.展开更多
In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equa...In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton＇s principle. Finally, the Hamilton＇s principle of high-order Lagrangian function is given.展开更多
Objective Repetitive transcranial magnetic stimulation(rTMS)has demonstrated efficacy in enhancing neurocognitive performance in Alzheimer’s disease(AD),but the neurobiological mechanisms linking synaptic pathology,n...Objective Repetitive transcranial magnetic stimulation(rTMS)has demonstrated efficacy in enhancing neurocognitive performance in Alzheimer’s disease(AD),but the neurobiological mechanisms linking synaptic pathology,neural oscillatory dynamics,and brain network reorganization remain unclear.This investigation seeks to systematically evaluate the therapeutic potential of rTMS as a non-invasive neuromodulatory intervention through a multimodal framework integrating clinical assessments,molecular profiling,and neurophysiological monitoring.Methods In this prospective double-blind trial,12 AD patients underwent a 14-day protocol of 20 Hz rTMS,with comprehensive multimodal assessments performed pre-and postintervention.Cognitive functioning was quantified using the mini-mental state examination(MMSE)and Montreal cognitive assessment(MOCA),while daily living capacities and neuropsychiatric profiles were respectively evaluated through the activities of daily living(ADL)scale and combined neuropsychiatric inventory(NPI)-Hamilton depression rating scale(HAMD).Peripheral blood biomarkers,specifically Aβ1-40 and phosphorylated tau(p-tau181),were analyzed to investigate the effects of rTMS on molecular metabolism.Spectral power analysis was employed to investigate rTMS-induced modulations of neural rhythms in AD patients,while brain network analyses incorporating topological properties were conducted to examine stimulus-driven network reorganization.Furthermore,systematic assessment of correlations between cognitive scale scores,blood biomarkers,and network characteristics was performed to elucidate cross-modal therapeutic associations.Results Clinically,MMSE and MOCA scores improved significantly(P<0.05).Biomarker showed that Aβ1-40 level increased(P<0.05),contrasting with p-tau181 reduction.Moreover,the levels of Aβ1-40 were positively correlated with MMSE and MOCA scores.Post-intervention analyses revealed significant modulations in oscillatory power,characterized by pronounced reductions in delta(P<0.05)and theta bands(P<0.05),while concurrent enhancements were observed in alpha,beta,and gamma band activities(all P<0.05).Network analysis revealed frequency-specific reorganization:clustering coefficients were significantly decreased in delta,theta,and alpha bands(P<0.05),while global efficiency improvement was exclusively detected in the delta band(P<0.05).The alpha band demonstrated concurrent increases in average nodal degree(P<0.05)and characteristic path length reduction(P<0.05).Further research findings indicate that the changes in the clinical scale HAMD scores before and after rTMS stimulation are negatively correlated with the changes in the blood biomarkers Aβ1-40 and p-tau181.Additionally,the changes in the clinical scales MMSE and MoCA scores were negatively correlated with the changes in the node degree of the alpha frequency band and negatively correlated with the clustering coefficient of the delta frequency band.However,the changes in MMSE scores are positively correlated with the changes in global efficiency of both the delta and alpha frequency bands.Conclusion 20 Hz rTMS targeting dorsolateral prefrontal cortex(DLPFC)significantly improves cognitive function and enhances the metabolic clearance ofβ-amyloid and tau proteins in AD patients.This neurotherapeutic effect is mechanistically associated with rTMS-mediated frequency-selective neuromodulation,which enhances the connectivity of oscillatory networks through improved neuronal synchronization and optimized topological organization of functional brain networks.These findings not only support the efficacy of rTMS as an adjunctive therapy for AD but also underscore the importance of employing multiple assessment methods—including clinical scales,blood biomarkers,and EEG——in understanding and monitoring the progression of AD.This research provides a significant theoretical foundation and empirical evidence for further exploration of rTMS applications in AD treatment.展开更多
Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order...Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. Spectral collocation method based on Lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.展开更多
In this paper,we obtain a normality criterion for families of meromorphic functions concerning‘wandering’shared functions,which generalizes and improves Montel’s criterion and the related results due to Schwick,Xu-...In this paper,we obtain a normality criterion for families of meromorphic functions concerning‘wandering’shared functions,which generalizes and improves Montel’s criterion and the related results due to Schwick,Xu-Fang,Xu-Qiu,and Grahl-Nevo.Also,a normality relationship between two families is given.展开更多
In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the wellknown Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions an...In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the wellknown Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether＇s theorem is presented. An example was discussed to illustrate these results.展开更多
Defect engineering is a commonly methodology used to enhance the thermoelectric performance of thermoelectric PbTe by improving its electronic transport properties.At the nanoscale,defects can induce quantum tunneling...Defect engineering is a commonly methodology used to enhance the thermoelectric performance of thermoelectric PbTe by improving its electronic transport properties.At the nanoscale,defects can induce quantum tunneling effects that significantly impact the electrical properties of materials.To understand the specific mechanisms underlying the quantum transport properties of PbTe,we employ the non-equilibrium Green's function(NEGF)method to investigate the effects of intrinsic defects(point defects and grain boundaries)on the electronic transport properties of PbTe-based nanodevices from a quantum mechanical perspective.Our results show that the Pb vacancy(VPb)has the highest conduction.The conduction depends on the defect type,chemical potential and bias voltage.The presence of intrinsic point defects introduces impurity levels,facilitating the electron tunneling and leading to an increase in the transmission coefficient,thereby enhancing the electronic transport properties.For PbTe containing grain boundaries,these boundaries suppress the electronic transport properties.The Te occupied twin boundary(Te-TB)exerts a stronger inhibitory effect than the Pb occupied twin boundary(Pb-TB).Nevertheless,the combined effect between twin boundaries and point defects can enhance the electrical properties.Therefore,in order to obtain highly conductive of PbTe materials,a Te-rich synthesis environment should be used to promote the effective formation of Pb vacancy.Our work offers a comprehensive understanding of the impact of defects on electron scattering in thermoelectric materials.展开更多
In this paper,we study a class of Sturm-Liouville problems,where the boundary conditions involve eigenparameters.Firstly,by defining a new inner product which depends on the transmission conditions,we obtain a new Hil...In this paper,we study a class of Sturm-Liouville problems,where the boundary conditions involve eigenparameters.Firstly,by defining a new inner product which depends on the transmission conditions,we obtain a new Hilbert space,on which the concerned operator A is self-adjoint.Then we construct the fundamental solutions to the problem,obtain the necessary and sufficient conditions for eigenvalues,and prove that the eigenvalues are simple.Finally,we investigate Green’s functions of such problem.展开更多
Understanding the effects of point liquid loading on transversely isotropic poroelastic media is crucial for advancing geomechanics and biomechanics, where precise modeling of fluid-structure interactions is essential...Understanding the effects of point liquid loading on transversely isotropic poroelastic media is crucial for advancing geomechanics and biomechanics, where precise modeling of fluid-structure interactions is essential. This paper presents a comprehensive analysis of infinite transversely isotropic poroelasticity under a fluid source, based on Biot's theory, aiming to uncover new and previously unexplored insights in the literature. We begin our study by deriving a general solution for fluid-saturated, transversely isotropic poroelastic materials in terms of harmonic functions that satisfy sixth-order homogeneous partial differential equations, using potential theory and Almansi's theorem. Based on these general solutions and potential functions, we construct a Green's function for a point fluid source, introducing three new harmonic functions with undetermined constants. These constants are determined by enforcing continuity and equilibrium conditions. Substituting these into the general solution yields fundamental solutions for poroelasticity that provide crucial support for a wide range of project problems. Numerical results and comparisons with existing literature are provided to illustrate physical mechanisms through contour plots. Our observations reveal that all components tend to zero in the far field and become singular at the concentrated source. Additionally, the contours exhibit rapid changes near the point fluid source but display gradual variations at a distance from it. These findings highlight the intricate behavior of the system under point liquid loading, offering valuable insights for further research and practical applications.展开更多
基金The second named author was supported in part by an NSERC Postdoctoral Fellowship,Canada and a CR F Grant,University of Alberta
文摘This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).
文摘This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.
基金Supported by the National Natural Science Foundation of China(Grant No.11871006).
文摘This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation polynomials based on the zeros of polynomials with the leading coefficient 1 of the least deviation from zero in L_(p,ω)[-1,1]are optimal for 1≤p<∞.We also give the optimal Lagrange interpolation nodes when the endpoints are included in the nodes.
文摘This paper deals with the description and the representation of polynomials defined over n-simplices, The polynomials are computed by using two recurrent schemes: the Neville-Aitken one for the Lagrange interpolating operator and the De Casteljau one for the Bernstein-Bezier approximating operator. Both schemes fall intothe framework of transformations of the form where the F iare given numbers (forexample, at the initial step they coincide with the values of the function on a given lattice), and the coefficients (x) are linear polynomials valued in x and x is fixed. A general theory for such sequence of transformations can be found in [2] where it is also proved that these tranformations are completely characterized in term of a linear functional, reference functional. This functional is associated with a linear space., characteristic space.The concepts of reference functionals and characteristic spaces will be used and we shall prove the existence of a characteristic space for the reference functional: associated with these operators.
文摘为提高网络入侵检测系统中检测算法的分类精度,降低训练样本及学习时间,在基于支持向量回归机的基础上,提出一种新的利用Lagrange支持向量回归机设计IDS的检测算法。使用KDD CUP 1999数据集进行仿真实验,结果表明该算法较基于支持向量回归机的检测算法具有更良好的泛化性能、更快的迭代速度、更高的检测精度和更低的误报率。
文摘Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.
基金the Natural Science Foundation of Jiangxi Provincethe Foundation of Education Department of Jiangxi Province under Grant No.[2007]136
文摘In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton＇s principle. Finally, the Hamilton＇s principle of high-order Lagrangian function is given.
文摘Objective Repetitive transcranial magnetic stimulation(rTMS)has demonstrated efficacy in enhancing neurocognitive performance in Alzheimer’s disease(AD),but the neurobiological mechanisms linking synaptic pathology,neural oscillatory dynamics,and brain network reorganization remain unclear.This investigation seeks to systematically evaluate the therapeutic potential of rTMS as a non-invasive neuromodulatory intervention through a multimodal framework integrating clinical assessments,molecular profiling,and neurophysiological monitoring.Methods In this prospective double-blind trial,12 AD patients underwent a 14-day protocol of 20 Hz rTMS,with comprehensive multimodal assessments performed pre-and postintervention.Cognitive functioning was quantified using the mini-mental state examination(MMSE)and Montreal cognitive assessment(MOCA),while daily living capacities and neuropsychiatric profiles were respectively evaluated through the activities of daily living(ADL)scale and combined neuropsychiatric inventory(NPI)-Hamilton depression rating scale(HAMD).Peripheral blood biomarkers,specifically Aβ1-40 and phosphorylated tau(p-tau181),were analyzed to investigate the effects of rTMS on molecular metabolism.Spectral power analysis was employed to investigate rTMS-induced modulations of neural rhythms in AD patients,while brain network analyses incorporating topological properties were conducted to examine stimulus-driven network reorganization.Furthermore,systematic assessment of correlations between cognitive scale scores,blood biomarkers,and network characteristics was performed to elucidate cross-modal therapeutic associations.Results Clinically,MMSE and MOCA scores improved significantly(P<0.05).Biomarker showed that Aβ1-40 level increased(P<0.05),contrasting with p-tau181 reduction.Moreover,the levels of Aβ1-40 were positively correlated with MMSE and MOCA scores.Post-intervention analyses revealed significant modulations in oscillatory power,characterized by pronounced reductions in delta(P<0.05)and theta bands(P<0.05),while concurrent enhancements were observed in alpha,beta,and gamma band activities(all P<0.05).Network analysis revealed frequency-specific reorganization:clustering coefficients were significantly decreased in delta,theta,and alpha bands(P<0.05),while global efficiency improvement was exclusively detected in the delta band(P<0.05).The alpha band demonstrated concurrent increases in average nodal degree(P<0.05)and characteristic path length reduction(P<0.05).Further research findings indicate that the changes in the clinical scale HAMD scores before and after rTMS stimulation are negatively correlated with the changes in the blood biomarkers Aβ1-40 and p-tau181.Additionally,the changes in the clinical scales MMSE and MoCA scores were negatively correlated with the changes in the node degree of the alpha frequency band and negatively correlated with the clustering coefficient of the delta frequency band.However,the changes in MMSE scores are positively correlated with the changes in global efficiency of both the delta and alpha frequency bands.Conclusion 20 Hz rTMS targeting dorsolateral prefrontal cortex(DLPFC)significantly improves cognitive function and enhances the metabolic clearance ofβ-amyloid and tau proteins in AD patients.This neurotherapeutic effect is mechanistically associated with rTMS-mediated frequency-selective neuromodulation,which enhances the connectivity of oscillatory networks through improved neuronal synchronization and optimized topological organization of functional brain networks.These findings not only support the efficacy of rTMS as an adjunctive therapy for AD but also underscore the importance of employing multiple assessment methods—including clinical scales,blood biomarkers,and EEG——in understanding and monitoring the progression of AD.This research provides a significant theoretical foundation and empirical evidence for further exploration of rTMS applications in AD treatment.
文摘Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. Spectral collocation method based on Lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.
基金Supported by the National Natural Science Foundation of China(Grant No.11471163)。
文摘In this paper,we obtain a normality criterion for families of meromorphic functions concerning‘wandering’shared functions,which generalizes and improves Montel’s criterion and the related results due to Schwick,Xu-Fang,Xu-Qiu,and Grahl-Nevo.Also,a normality relationship between two families is given.
基金Project supported by the National Natural Science Foundation of China(Grant No.10872037)the Natural Science Foundationof Anhui Province,China(Grant No.070416226)
文摘In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the wellknown Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether＇s theorem is presented. An example was discussed to illustrate these results.
基金financial support from the National Natural Science Foundation of China(No.12474016)the program of“Distinguished Expert of Taishan Scholar”(No.tstp20221124)+4 种基金the National Natural Science Foundation of China(Nos.52172212,12474017)the Shandong Provincial Science Foundation(ZR2021YQ03)the program for“Young Scientists of Taishan Scholars”(No.tsqn202306184)financial support from the National Natural Science Foundation of China(No.12464034)the Natural Science Foundation of Ningxia,China(No.2024AAC05070)。
文摘Defect engineering is a commonly methodology used to enhance the thermoelectric performance of thermoelectric PbTe by improving its electronic transport properties.At the nanoscale,defects can induce quantum tunneling effects that significantly impact the electrical properties of materials.To understand the specific mechanisms underlying the quantum transport properties of PbTe,we employ the non-equilibrium Green's function(NEGF)method to investigate the effects of intrinsic defects(point defects and grain boundaries)on the electronic transport properties of PbTe-based nanodevices from a quantum mechanical perspective.Our results show that the Pb vacancy(VPb)has the highest conduction.The conduction depends on the defect type,chemical potential and bias voltage.The presence of intrinsic point defects introduces impurity levels,facilitating the electron tunneling and leading to an increase in the transmission coefficient,thereby enhancing the electronic transport properties.For PbTe containing grain boundaries,these boundaries suppress the electronic transport properties.The Te occupied twin boundary(Te-TB)exerts a stronger inhibitory effect than the Pb occupied twin boundary(Pb-TB).Nevertheless,the combined effect between twin boundaries and point defects can enhance the electrical properties.Therefore,in order to obtain highly conductive of PbTe materials,a Te-rich synthesis environment should be used to promote the effective formation of Pb vacancy.Our work offers a comprehensive understanding of the impact of defects on electron scattering in thermoelectric materials.
基金supported by the National Natural Science Foundation of China(No.12461086)the Natural Science Foundation of Hubei Province(No.2022CFC016)。
文摘In this paper,we study a class of Sturm-Liouville problems,where the boundary conditions involve eigenparameters.Firstly,by defining a new inner product which depends on the transmission conditions,we obtain a new Hilbert space,on which the concerned operator A is self-adjoint.Then we construct the fundamental solutions to the problem,obtain the necessary and sufficient conditions for eigenvalues,and prove that the eigenvalues are simple.Finally,we investigate Green’s functions of such problem.
基金supported by the National Natural Science Foundation of China (Grant Nos. 12272269, 11972257,11832014 and 11472193)the Shanghai Pilot Program for Basic Researchthe Shanghai Gaofeng Project for University Academic Program Development。
文摘Understanding the effects of point liquid loading on transversely isotropic poroelastic media is crucial for advancing geomechanics and biomechanics, where precise modeling of fluid-structure interactions is essential. This paper presents a comprehensive analysis of infinite transversely isotropic poroelasticity under a fluid source, based on Biot's theory, aiming to uncover new and previously unexplored insights in the literature. We begin our study by deriving a general solution for fluid-saturated, transversely isotropic poroelastic materials in terms of harmonic functions that satisfy sixth-order homogeneous partial differential equations, using potential theory and Almansi's theorem. Based on these general solutions and potential functions, we construct a Green's function for a point fluid source, introducing three new harmonic functions with undetermined constants. These constants are determined by enforcing continuity and equilibrium conditions. Substituting these into the general solution yields fundamental solutions for poroelasticity that provide crucial support for a wide range of project problems. Numerical results and comparisons with existing literature are provided to illustrate physical mechanisms through contour plots. Our observations reveal that all components tend to zero in the far field and become singular at the concentrated source. Additionally, the contours exhibit rapid changes near the point fluid source but display gradual variations at a distance from it. These findings highlight the intricate behavior of the system under point liquid loading, offering valuable insights for further research and practical applications.
