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
Generally the incompressible viscous flow problem is described by the Navier-Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the s...Generally the incompressible viscous flow problem is described by the Navier-Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the step-bystep computation in the field of time, and adopting the least-square estimation of the-same-order shift, this paper has calculated both velocity and pressure from the decoupling independent equations. Each time fraction Newton-Raphson iterative method is applied for the velocity and pressure. Finally, this paper puts the method into practice of the shear-drive cavity flow, verifying the validity, high accuracy and stability.展开更多
The weighted residuals method was used for obtaining the boundary integral representation of the velocity of the three-dimensional inviscid irrotational flow. It is shown that velocity in an arbitrary point of domain ...The weighted residuals method was used for obtaining the boundary integral representation of the velocity of the three-dimensional inviscid irrotational flow. It is shown that velocity in an arbitrary point of domain can be expressed through its values on the boundary. Boundary integral equations of the second kind for solving boundary-valued problems of the first and second kinds are developed. The result has been also generalised to the case of solenoidal vector fields with potential vorticity. It is shown that the resulting integral equations are Fredholm integral equations of the second kind and allow effective numerical solving of corresponding boundary-valued problems. Examples of numerical solutions for a sphere and an ellipsoid are given for demonstration of efficiency of the offered method.展开更多
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Generally the incompressible viscous flow problem is described by the Navier-Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the step-bystep computation in the field of time, and adopting the least-square estimation of the-same-order shift, this paper has calculated both velocity and pressure from the decoupling independent equations. Each time fraction Newton-Raphson iterative method is applied for the velocity and pressure. Finally, this paper puts the method into practice of the shear-drive cavity flow, verifying the validity, high accuracy and stability.
文摘The weighted residuals method was used for obtaining the boundary integral representation of the velocity of the three-dimensional inviscid irrotational flow. It is shown that velocity in an arbitrary point of domain can be expressed through its values on the boundary. Boundary integral equations of the second kind for solving boundary-valued problems of the first and second kinds are developed. The result has been also generalised to the case of solenoidal vector fields with potential vorticity. It is shown that the resulting integral equations are Fredholm integral equations of the second kind and allow effective numerical solving of corresponding boundary-valued problems. Examples of numerical solutions for a sphere and an ellipsoid are given for demonstration of efficiency of the offered method.
