The existence of positive solutions to second-order periodic BVPs-u'+Mu =j(t, u),t(0) = u(2π),u'(0) = '(2π) and u'+ Mu = I(t, u), u(0) = u(2π), u'(0) = u'(2π)is proved by a simple appliCati...The existence of positive solutions to second-order periodic BVPs-u'+Mu =j(t, u),t(0) = u(2π),u'(0) = '(2π) and u'+ Mu = I(t, u), u(0) = u(2π), u'(0) = u'(2π)is proved by a simple appliCation of a Fixed point Theorem in cones due to Krasnoselskii.展开更多
This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential o...This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.展开更多
The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and ...The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.展开更多
The effects of a magnetic dipole on a nonlinear thermally radiative ferromagnetic liquidflowing over a stretched surface in the presence of Brownian motion and thermophoresis are investigated.By means of a similarity t...The effects of a magnetic dipole on a nonlinear thermally radiative ferromagnetic liquidflowing over a stretched surface in the presence of Brownian motion and thermophoresis are investigated.By means of a similarity transformation,ordinary differential equations are derived and solved afterwards using a numerical(the BVP4C)method.The impact of various parameters,namely the velocity,temperature,concentration,is presented graphically.It is shown that the nanoparticles properties,in conjunction with the magnetic dipole effect,can increase the thermal conductivity of the engineered nanofluid and,consequently,the heat transfer.Comparison with earlier studies indicates high accuracy and effectiveness of the numerical approach.An increase in the Brow-nian motion parameter and thermophoresis parameter enhances the concentration and the related boundary layer.The skin-friction rises when the viscosity parameter is increased.A larger value of the ferromagnetic para-meter results in a higher skin-friction and,vice versa,in a smaller Nusselt number.展开更多
Motivated by the widespread applications of nanofluids,a nanofluid model is proposed which focuses on uniform magnetohydrodynamic(MHD)boundary layer flow over a non-linear stretching sheet,incorporating the Casson mod...Motivated by the widespread applications of nanofluids,a nanofluid model is proposed which focuses on uniform magnetohydrodynamic(MHD)boundary layer flow over a non-linear stretching sheet,incorporating the Casson model for blood-based nanofluid while accounting for viscous and Ohmic dissipation effects under the cases of Constant Surface Temperature(CST)and Prescribed Surface Temperature(PST).The study employs a twophase model for the nanofluid,coupled with thermophoresis and Brownian motion,to analyze the effects of key fluid parameters such as thermophoresis,Brownian motion,slip velocity,Schmidt number,Eckert number,magnetic parameter,and non-linear stretching parameter on the velocity,concentration,and temperature profiles of the nanofluid.The proposed model is novel as it simultaneously considers the impact of thermophoresis and Brownian motion,along with Ohmic and viscous dissipation effects,in both CST and PST scenarios for blood-based Casson nanofluid.The numerical technique built into MATLAB’s bvp4c module is utilized to solve the governing system of coupled differential equations,revealing that the concentration of nanoparticles decreases with increasing thermophoresis and Brownian motion parameters while the temperature of the nanofluid increases.Additionally,a higher Eckert number is found to reduce the nanofluid temperature.A comparative analysis between CST and PST scenarios is also undertaken,which highlights the significant influence of these factors on the fluid’s characteristics.The findings have potential applications in biomedical processes to enhance fluid velocity and heat transfer rates,ultimately improving patient outcomes.展开更多
This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We der...This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in detail, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory. .展开更多
In this article, the author employs the conical expansion and compression fixed point principle and the fixed point index theory to show that there exist at least two positive solutions for a higher order BVP.
The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and...The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.展开更多
The purpose of this research is to investigate the influence that slip boundary conditions have on the rate of heat and mass transfer by examining the behavior of micropolar MHD flow across a porous stretching sheet.I...The purpose of this research is to investigate the influence that slip boundary conditions have on the rate of heat and mass transfer by examining the behavior of micropolar MHD flow across a porous stretching sheet.In addition to this,the impacts of thermal radiation and viscous dissipation are taken into account.With the use of various computing strategies,numerical results have been produced.Similarity transformation was utilized in order to convert the partial differential equations(PDEs)that regulated energy,rotational momentum,concentration,and momentum into ordinary differential equations(ODEs).As compared to earlier published research,MATLAB inbuilt solver solution shows an extremely good correlation in exceptional instances.In exceptional instances,the present MATLAB inbuilt solver solution has a very excellent connection with the findings of the previously published investigations.A variety of flow field factors impact the Nusselt number,the wall couple shear stress,the friction factor,Sherwood numbers the dimensionless distributions discussed in detail.When the Eckert number rises,the temperature rises,and the Schmidt number falls,the concentration falls.Velocity increases with increases in the material factor but drops with increases in the magnetic parameter and the surface condition factor.展开更多
The present study numerically investigates the flow and heat transfer of porous Williamson hybrid nanofluid on an exponentially shrinking sheet with magnetohydrodynamic(MHD)effects.The nonlinear partial differential e...The present study numerically investigates the flow and heat transfer of porous Williamson hybrid nanofluid on an exponentially shrinking sheet with magnetohydrodynamic(MHD)effects.The nonlinear partial differential equations which governed the model are first reduced to a set of ordinary differential equations by using the similarity transformation.Next,the BVP4C solver is applied to solve the equations by considering the pertinent fluid parameters such as the permeability parameter,the magnetic parameter,the Williamson parameter,the nanoparticle volume fractions and the wall mass transfer parameter.The single(SWCNTs)and multi-walled carbon nanotubes(MWCNTs)nanoparticles are taken as the hybrid nanoparticles.It is found that the increase in magnetic parameter in SWCNT+MCWNT hybrid nanofluid results in an increase of 72.2%on skin friction compared to SWCNT nanofluid while maintaining reducing a small number of Nusselt number.This shows the potential of the Williamson hybrid nanofluid for friction application purposes especially in transportation like braking system,flushing fluid and mechanical engineering.展开更多
The boundary value problems of the third-order ordinary differential equation have many practical application backgrounds and their some special cases have been studied by many authors. However, few scholars have stud...The boundary value problems of the third-order ordinary differential equation have many practical application backgrounds and their some special cases have been studied by many authors. However, few scholars have studied the boundary value problems of the complete third-order differential equations u′′′(t) = f (t,u(t),u′(t),u′′(t)). In this paper, we discuss the existence and uniqueness of solutions and positive solutions of the fully third-order ordinary differential equation on [0,1] with the boundary condition u(0) = u′(1) = u′′(1) = 0. Under some inequality conditions on nonlinearity f some new existence and uniqueness results of solutions and positive solutions are obtained.展开更多
文摘The existence of positive solutions to second-order periodic BVPs-u'+Mu =j(t, u),t(0) = u(2π),u'(0) = '(2π) and u'+ Mu = I(t, u), u(0) = u(2π), u'(0) = u'(2π)is proved by a simple appliCation of a Fixed point Theorem in cones due to Krasnoselskii.
文摘This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates.
文摘The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.
文摘The effects of a magnetic dipole on a nonlinear thermally radiative ferromagnetic liquidflowing over a stretched surface in the presence of Brownian motion and thermophoresis are investigated.By means of a similarity transformation,ordinary differential equations are derived and solved afterwards using a numerical(the BVP4C)method.The impact of various parameters,namely the velocity,temperature,concentration,is presented graphically.It is shown that the nanoparticles properties,in conjunction with the magnetic dipole effect,can increase the thermal conductivity of the engineered nanofluid and,consequently,the heat transfer.Comparison with earlier studies indicates high accuracy and effectiveness of the numerical approach.An increase in the Brow-nian motion parameter and thermophoresis parameter enhances the concentration and the related boundary layer.The skin-friction rises when the viscosity parameter is increased.A larger value of the ferromagnetic para-meter results in a higher skin-friction and,vice versa,in a smaller Nusselt number.
基金funded by Universiti Teknikal Malaysia Melaka and Ministry of Higher Education(MoHE)Malaysia,grant number FRGS/1/2024/FTKM/F00586.
文摘Motivated by the widespread applications of nanofluids,a nanofluid model is proposed which focuses on uniform magnetohydrodynamic(MHD)boundary layer flow over a non-linear stretching sheet,incorporating the Casson model for blood-based nanofluid while accounting for viscous and Ohmic dissipation effects under the cases of Constant Surface Temperature(CST)and Prescribed Surface Temperature(PST).The study employs a twophase model for the nanofluid,coupled with thermophoresis and Brownian motion,to analyze the effects of key fluid parameters such as thermophoresis,Brownian motion,slip velocity,Schmidt number,Eckert number,magnetic parameter,and non-linear stretching parameter on the velocity,concentration,and temperature profiles of the nanofluid.The proposed model is novel as it simultaneously considers the impact of thermophoresis and Brownian motion,along with Ohmic and viscous dissipation effects,in both CST and PST scenarios for blood-based Casson nanofluid.The numerical technique built into MATLAB’s bvp4c module is utilized to solve the governing system of coupled differential equations,revealing that the concentration of nanoparticles decreases with increasing thermophoresis and Brownian motion parameters while the temperature of the nanofluid increases.Additionally,a higher Eckert number is found to reduce the nanofluid temperature.A comparative analysis between CST and PST scenarios is also undertaken,which highlights the significant influence of these factors on the fluid’s characteristics.The findings have potential applications in biomedical processes to enhance fluid velocity and heat transfer rates,ultimately improving patient outcomes.
文摘This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in detail, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory. .
基金the National Natural Science Foundation of China (No.19671052) and Postdoctorate Foundation of Zhengzhou Uniyersity.
文摘In this article, the author employs the conical expansion and compression fixed point principle and the fixed point index theory to show that there exist at least two positive solutions for a higher order BVP.
文摘The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.
文摘The purpose of this research is to investigate the influence that slip boundary conditions have on the rate of heat and mass transfer by examining the behavior of micropolar MHD flow across a porous stretching sheet.In addition to this,the impacts of thermal radiation and viscous dissipation are taken into account.With the use of various computing strategies,numerical results have been produced.Similarity transformation was utilized in order to convert the partial differential equations(PDEs)that regulated energy,rotational momentum,concentration,and momentum into ordinary differential equations(ODEs).As compared to earlier published research,MATLAB inbuilt solver solution shows an extremely good correlation in exceptional instances.In exceptional instances,the present MATLAB inbuilt solver solution has a very excellent connection with the findings of the previously published investigations.A variety of flow field factors impact the Nusselt number,the wall couple shear stress,the friction factor,Sherwood numbers the dimensionless distributions discussed in detail.When the Eckert number rises,the temperature rises,and the Schmidt number falls,the concentration falls.Velocity increases with increases in the material factor but drops with increases in the magnetic parameter and the surface condition factor.
文摘The present study numerically investigates the flow and heat transfer of porous Williamson hybrid nanofluid on an exponentially shrinking sheet with magnetohydrodynamic(MHD)effects.The nonlinear partial differential equations which governed the model are first reduced to a set of ordinary differential equations by using the similarity transformation.Next,the BVP4C solver is applied to solve the equations by considering the pertinent fluid parameters such as the permeability parameter,the magnetic parameter,the Williamson parameter,the nanoparticle volume fractions and the wall mass transfer parameter.The single(SWCNTs)and multi-walled carbon nanotubes(MWCNTs)nanoparticles are taken as the hybrid nanoparticles.It is found that the increase in magnetic parameter in SWCNT+MCWNT hybrid nanofluid results in an increase of 72.2%on skin friction compared to SWCNT nanofluid while maintaining reducing a small number of Nusselt number.This shows the potential of the Williamson hybrid nanofluid for friction application purposes especially in transportation like braking system,flushing fluid and mechanical engineering.
文摘The boundary value problems of the third-order ordinary differential equation have many practical application backgrounds and their some special cases have been studied by many authors. However, few scholars have studied the boundary value problems of the complete third-order differential equations u′′′(t) = f (t,u(t),u′(t),u′′(t)). In this paper, we discuss the existence and uniqueness of solutions and positive solutions of the fully third-order ordinary differential equation on [0,1] with the boundary condition u(0) = u′(1) = u′′(1) = 0. Under some inequality conditions on nonlinearity f some new existence and uniqueness results of solutions and positive solutions are obtained.
