We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We ob...We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.展开更多
In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be r...In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.展开更多
By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is propose...By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.展开更多
In this paper, we first obtain some new results about the existence of multiple positive solutions for singular impulsive boundary value problems, and then to illustrate our main results we studied the existence of mu...In this paper, we first obtain some new results about the existence of multiple positive solutions for singular impulsive boundary value problems, and then to illustrate our main results we studied the existence of multiple positive solutions for an infinite system of scalar equations.展开更多
This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫...This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫∫_(R2N)|x|^(α1p)|y|^(α2p)|u(x) − u(y)|^(p)/|x − y|^(N+ps) dxdy)L_(p)^(s)u = |x| ^(β)f(u) in Ω,u = 0 in R^(N) \ Ω,where L_(p)^(s) is the generalized fractional p-Laplacian operator,N≥1,s∈(0,1),α_(1),α_(2),β∈R,Ω■R^(N) is a bounded domain with Lipschitz boundary,and M:R0^(+)→R0^(+),f:Ω→R are continuous functions.Firstly,we introduce a variational framework for the above problem.Then,the existence of least energy solutions is obtained by using variational methods,provided that the nonlinear term f has(θ_(p-1))-sublinear growth at infinity.Moreover,the existence of infinitely many solutions is obtained by using Krasnoselskii’s genus theory.Finally,we obtain the existence and multiplicity of solutions if f has(θ_(p-1))-superlinear growth at infinity.The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.展开更多
A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems i...A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.展开更多
This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2...This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.展开更多
This paper deals with the existence of positive solutions to the singular boundary value problemwhere q(t) may be singular at t = 0 and t = 1, f(t,y) may be superlinear at y =∞ and singular, at y = 0.
In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = ...In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.展开更多
New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the cond...New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.展开更多
In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error esti...In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.展开更多
This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matr...This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.展开更多
Singular initial value problems arise in solving one-dimensional steady transonic flow of dualmode scramjet. The existing solution method has the problems of large initial value errors in principles. This paper puts f...Singular initial value problems arise in solving one-dimensional steady transonic flow of dualmode scramjet. The existing solution method has the problems of large initial value errors in principles. This paper puts forward an improved algorithm based on variable transformation, and constructs a nonsingular one-dimensional steady transonic flow equation by defining a new variable. The improved algorithm can eliminate the singularity of the differential equation, and can solve the singular initial value problems of one-dimensional steady transonic flow of dual-mode scramjet.展开更多
In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theo...In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.展开更多
The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching...The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching principle.展开更多
An efficient analytical decomposition technique was presented for solving the singular nonlinear boundary value problem arising in viscous flow when the Crocco variable was introduced. The approximate analytical solut...An efficient analytical decomposition technique was presented for solving the singular nonlinear boundary value problem arising in viscous flow when the Crocco variable was introduced. The approximate analytical solution may be represented in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of the approximate solutions were verified by numerical ones in the literature. The approximate analytical solutions can be successfully applied to give the values of skin friction coefficient.展开更多
The existence of at least two positive solutions is presented for the singular second-order boundary value problem{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1, limt→0 p（t）x′（t）=...The existence of at least two positive solutions is presented for the singular second-order boundary value problem{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1, limt→0 p（t）x′（t）=0,x（1）=0by using the fixed point index, where f may be singular at x = 0 and px ′= 0.展开更多
We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） ...We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） quenches at the originx = 0 at the same time ＇1＇ （see Theorem 4.3）. We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.展开更多
The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered.Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of...The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered.Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.展开更多
In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the met...In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.展开更多
基金supported by Shandong Provincial NSF(ZR2022MA020).
文摘We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.
基金supported by the National Natural Science Foundation of China (No.12172154)the 111 Project (No.B14044)+1 种基金the Natural Science Foundation of Gansu Province (No.23JRRA1035)the Natural Science Foundation of Anhui University of Finance and Economics (No.ACKYC20043).
文摘In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.
基金supported by the Natural Science Foundation of Zhejiang Province,China(Grant Nos.LY20A010021,LY19A010002,LY20G030025)the Natural Science Founda-tion of Ningbo City,China(Grant Nos.2021J147,2021J235).
文摘By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.
基金The NSF (01BXL002) of Xuzhou Normal University and the NSF (03KJB110137) of Jingsu Education Committee.
文摘In this paper, we first obtain some new results about the existence of multiple positive solutions for singular impulsive boundary value problems, and then to illustrate our main results we studied the existence of multiple positive solutions for an infinite system of scalar equations.
基金supported by National Natural Science Foundation of China(11601515)Fundamental Research Funds for the Central Universities(3122017080)+3 种基金the second author acknowledges the support of the Slovenian Research Agency grants P1-0292,J1-8131,N1-0064,N1-0083,N1-0114the third author was supported by National Natural Science Foundation of China(11871199and 12171152)Shandong Provincial Natural Science Foundation,PR China(ZR2020MA006)Cultivation Project of Young and Innovative Talents in Universities of Shandong Province。
文摘This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫∫_(R2N)|x|^(α1p)|y|^(α2p)|u(x) − u(y)|^(p)/|x − y|^(N+ps) dxdy)L_(p)^(s)u = |x| ^(β)f(u) in Ω,u = 0 in R^(N) \ Ω,where L_(p)^(s) is the generalized fractional p-Laplacian operator,N≥1,s∈(0,1),α_(1),α_(2),β∈R,Ω■R^(N) is a bounded domain with Lipschitz boundary,and M:R0^(+)→R0^(+),f:Ω→R are continuous functions.Firstly,we introduce a variational framework for the above problem.Then,the existence of least energy solutions is obtained by using variational methods,provided that the nonlinear term f has(θ_(p-1))-sublinear growth at infinity.Moreover,the existence of infinitely many solutions is obtained by using Krasnoselskii’s genus theory.Finally,we obtain the existence and multiplicity of solutions if f has(θ_(p-1))-superlinear growth at infinity.The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.
文摘A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.
文摘This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.
文摘This paper deals with the existence of positive solutions to the singular boundary value problemwhere q(t) may be singular at t = 0 and t = 1, f(t,y) may be superlinear at y =∞ and singular, at y = 0.
基金supported by the National Natural Science Foundation of China (11071149, 10771128)the NSF of Shanxi Province (2006011002, 2010011001-1)
文摘In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem： -u′′（t） = λf（t, u（t）） for all t ∈ （0, 1） subjecting to u（0） = 0 and αu（η） = u（1）, where η ∈ （0, 1）, α ∈ [0, 1）, and λ is a positive parameter. The nonlinear term f（t, u） is nonnegative, and may be singular at t = 0, t = 1, and u = 0. By the fixed point index theory and approximation method, we establish that there exists λ＊ ∈ （0, ＋∞], such that the above problem has at least two positive solutions for any λ ∈ （0, λ＊） under certain conditions on the nonlinear term f.
文摘New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.
文摘In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.
基金Project supported by the National Natural Science Foundation of China(No.10672194)the China-Russia Cooperative Project(the National Natural Science Foundation of China and the Russian Foundation for Basic Research)(No.10811120012)
文摘This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.
基金Hi TechResearchandDevelopmentProgramofChina(2002AA723011),OutstandingYouthFoundationofHeilongjiang Province
文摘Singular initial value problems arise in solving one-dimensional steady transonic flow of dualmode scramjet. The existing solution method has the problems of large initial value errors in principles. This paper puts forward an improved algorithm based on variable transformation, and constructs a nonsingular one-dimensional steady transonic flow equation by defining a new variable. The improved algorithm can eliminate the singularity of the differential equation, and can solve the singular initial value problems of one-dimensional steady transonic flow of dual-mode scramjet.
基金supported by the Key Program of Scientific Research Fund for Young Teachers of AUST(QN2018109)the National Natural Science Foundation of China(11801008)+1 种基金supported by the Fundamental Research Funds for the Central Universities(2017B715X14)the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX17_0508)
文摘In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.
基金Supported by the NNSF of China（10901003） Supported by the Natural Science Foundation from the Education Bureau of Anhui Province（KJ2011A135）
文摘The existence and asymptotic behavior of solution for a class of quasilinear singularly perturbed boundary value problems are discussed under suitable conditions by the theory of differential inequalities and matching principle.
基金The work was financially supported by the National Natural Science Foundation of China (No.50476083) and the Cross-CenturyTalents Projects of the Educational Ministry of China.
文摘An efficient analytical decomposition technique was presented for solving the singular nonlinear boundary value problem arising in viscous flow when the Crocco variable was introduced. The approximate analytical solution may be represented in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of the approximate solutions were verified by numerical ones in the literature. The approximate analytical solutions can be successfully applied to give the values of skin friction coefficient.
基金the NNSFC(10571111)the Fundation of Natural Science of Shandong Province(Y2005A07)
文摘The existence of at least two positive solutions is presented for the singular second-order boundary value problem{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1, limt→0 p（t）x′（t）=0,x（1）=0by using the fixed point index, where f may be singular at x = 0 and px ′= 0.
基金supported by NSFC(11201380)the Fundamental Research Funds for the Central Universities(XDJK2012B007)+1 种基金Doctor Fund of Southwest University(SWU111021)Educational Fund of Southwest University(2010JY053)
文摘We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） quenches at the originx = 0 at the same time ＇1＇ （see Theorem 4.3）. We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.
文摘The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered.Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.
基金supported by the National Natural Science Foundation of China (11132004 and 51078145)the Natural Science Foundation of Guangdong Province (9251064101000016)
文摘In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.
