International Journal of Mathematics and Mathematical Sciences, 2006
We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points o... more We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points of generalized contractions.
International journal of pure and applied mathematics, 2015
Finite volume schemes for one dimensional Advection-Diffusion Equation (ADE) are discussed in thi... more Finite volume schemes for one dimensional Advection-Diffusion Equation (ADE) are discussed in this article. As a result, a general explicit difference equation of the form Un+1 m = aU n m−1+ bU n m+ cU n m+1 is obtained with general coefficients a, b, and c. Stability condition and local truncation error for this general form of explicit difference equation are derived. Then, total Variation Diminishing (TVD) schemes for general flux limiter ψ(r) are also discussed. Further, a relation between flux limiter and mesh length parameters is also obtained. Numerical justification for order of convergence for upwind, central difference and various TVD schemes are also presented. AMS Subject Classification: 65M08, 65M12, 65M15, 65N08, 65N12
A positivity preserving upwind scheme for one-dimensional species transport equation is discussed... more A positivity preserving upwind scheme for one-dimensional species transport equation is discussed in this article. It is proved that the proposed numerical scheme is unconditionally stable. Consistency of the scheme is also discussed in detail. It is shown that the local truncation error is consistent with the advection-diffusion-reaction equation when \(\varDelta t\rightarrow 0\) and inconsistent when \(\varDelta x\rightarrow 0\). Hence, the numerical approximation converges to exact solution only when \(\varDelta t\rightarrow 0\).
International Journal of Computational Methods, 2020
In this paper, numerical solutions of the extended Fisher–Kolmogorov equation are obtained using ... more In this paper, numerical solutions of the extended Fisher–Kolmogorov equation are obtained using finite pointset method. Finite pointset method is a meshless method which is a local iterative method based on the weighted least square approximation. By employing splitting technique, the extended Fisher–Kolmogorov equation is split into a two coupled system of differential equations by introducing an intermediate function. The method is applied to the resulting coupled system of differential equation. The numerical results confirm the good efficiency of the finite pointset method.
International Journal of Computational Methods, 2018
In this paper, a meshless method based on finite point set is presented for solving the biharmoni... more In this paper, a meshless method based on finite point set is presented for solving the biharmonic equation with simply supported boundary condition. The biharmonic equation is split into a coupled system of two Poisson equations by introducing an intermediate function. The system of two Poisson equations is then solved by finite pointset method. This method is a local iterative method based on the weighted least square approximation. The advantage of this method is that two resultant of sizes only [Formula: see text] matrices are solved at each particle for the original and intermediate solution. Numerical results indicate a good accuracy of the finite pointset method.
A quadrature based mixed Petrov-Galerkin finite element method is applied to a fourth order linea... more A quadrature based mixed Petrov-Galerkin finite element method is applied to a fourth order linear non-homogeneous ordinary differential equation with variable coefficients. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by Gauss quadrature rule in the formulation itself. Optimal order apriori error estimates in W k,p-norms for k = 0, 1, 2 and 1 ≤ p ≤ ∞ are obtained without any restriction on the mesh, not only for the approximation of the exact solution also for its second derivative. These error estimates are validated by a suitable numerical example.
An H 1 -Galerkin mixed finite element method is applied to the extended Fisher- Kolmogorov equati... more An H 1 -Galerkin mixed finite element method is applied to the extended Fisher- Kolmogorov equation by employing a splitting technique. The method described in this paper may also be considered as a Petrov-Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since second derivative of a cubic spline is a linear spline. Optimal order error estimates are obtained without any restriction on the mesh. Fully discrete scheme is also discussed and optimal order estimates are obtained. The results are validated with numerical examples.
International Journal of Mathematics and Mathematical Sciences, 2012
A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linea... more A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal ordera priorierror estimates are obtained without any restriction on the mesh.
Based on straightening the free boundary, a qualocation method is proposed and analysed for a sin... more Based on straightening the free boundary, a qualocation method is proposed and analysed for a single phase unidimensional Stefan problem. This method may be considered as a discrete version of the H 1 -Galerkin method in which the discretization is achieved by approximating the integrals by a composite Gauss quadrature rule. Optimal error estimates are derived in L ∞ (W j,∞ ), j = 0,1, and L ∞ (H j ), j = 0,1,2, norms for a semidiscrete scheme without any quasi-uniformity assumption on the finite element mesh.
International Journal of Mathematics and Mathematical Sciences, 2006
We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points o... more We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points of generalized contractions.
International journal of pure and applied mathematics, 2015
Finite volume schemes for one dimensional Advection-Diffusion Equation (ADE) are discussed in thi... more Finite volume schemes for one dimensional Advection-Diffusion Equation (ADE) are discussed in this article. As a result, a general explicit difference equation of the form Un+1 m = aU n m−1+ bU n m+ cU n m+1 is obtained with general coefficients a, b, and c. Stability condition and local truncation error for this general form of explicit difference equation are derived. Then, total Variation Diminishing (TVD) schemes for general flux limiter ψ(r) are also discussed. Further, a relation between flux limiter and mesh length parameters is also obtained. Numerical justification for order of convergence for upwind, central difference and various TVD schemes are also presented. AMS Subject Classification: 65M08, 65M12, 65M15, 65N08, 65N12
A positivity preserving upwind scheme for one-dimensional species transport equation is discussed... more A positivity preserving upwind scheme for one-dimensional species transport equation is discussed in this article. It is proved that the proposed numerical scheme is unconditionally stable. Consistency of the scheme is also discussed in detail. It is shown that the local truncation error is consistent with the advection-diffusion-reaction equation when \(\varDelta t\rightarrow 0\) and inconsistent when \(\varDelta x\rightarrow 0\). Hence, the numerical approximation converges to exact solution only when \(\varDelta t\rightarrow 0\).
International Journal of Computational Methods, 2020
In this paper, numerical solutions of the extended Fisher–Kolmogorov equation are obtained using ... more In this paper, numerical solutions of the extended Fisher–Kolmogorov equation are obtained using finite pointset method. Finite pointset method is a meshless method which is a local iterative method based on the weighted least square approximation. By employing splitting technique, the extended Fisher–Kolmogorov equation is split into a two coupled system of differential equations by introducing an intermediate function. The method is applied to the resulting coupled system of differential equation. The numerical results confirm the good efficiency of the finite pointset method.
International Journal of Computational Methods, 2018
In this paper, a meshless method based on finite point set is presented for solving the biharmoni... more In this paper, a meshless method based on finite point set is presented for solving the biharmonic equation with simply supported boundary condition. The biharmonic equation is split into a coupled system of two Poisson equations by introducing an intermediate function. The system of two Poisson equations is then solved by finite pointset method. This method is a local iterative method based on the weighted least square approximation. The advantage of this method is that two resultant of sizes only [Formula: see text] matrices are solved at each particle for the original and intermediate solution. Numerical results indicate a good accuracy of the finite pointset method.
A quadrature based mixed Petrov-Galerkin finite element method is applied to a fourth order linea... more A quadrature based mixed Petrov-Galerkin finite element method is applied to a fourth order linear non-homogeneous ordinary differential equation with variable coefficients. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by Gauss quadrature rule in the formulation itself. Optimal order apriori error estimates in W k,p-norms for k = 0, 1, 2 and 1 ≤ p ≤ ∞ are obtained without any restriction on the mesh, not only for the approximation of the exact solution also for its second derivative. These error estimates are validated by a suitable numerical example.
An H 1 -Galerkin mixed finite element method is applied to the extended Fisher- Kolmogorov equati... more An H 1 -Galerkin mixed finite element method is applied to the extended Fisher- Kolmogorov equation by employing a splitting technique. The method described in this paper may also be considered as a Petrov-Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since second derivative of a cubic spline is a linear spline. Optimal order error estimates are obtained without any restriction on the mesh. Fully discrete scheme is also discussed and optimal order estimates are obtained. The results are validated with numerical examples.
International Journal of Mathematics and Mathematical Sciences, 2012
A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linea... more A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal ordera priorierror estimates are obtained without any restriction on the mesh.
Based on straightening the free boundary, a qualocation method is proposed and analysed for a sin... more Based on straightening the free boundary, a qualocation method is proposed and analysed for a single phase unidimensional Stefan problem. This method may be considered as a discrete version of the H 1 -Galerkin method in which the discretization is achieved by approximating the integrals by a composite Gauss quadrature rule. Optimal error estimates are derived in L ∞ (W j,∞ ), j = 0,1, and L ∞ (H j ), j = 0,1,2, norms for a semidiscrete scheme without any quasi-uniformity assumption on the finite element mesh.
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