20 research outputs found
OPTIMAL APPROXIMATION SPACES FOR SOLVING PROBLEMS WITH ROUGH COEFFICIENTS
The finite element method has been widely used to solve partial
differential equations by both engineers and mathematicians for the
last several decades. This is due to its well-known effectiveness
when applied to a wide variety of problems. However, it has some
practical drawbacks. One of them is the need for meshing. Another is
that it uses polynomials as the approximation basis functions.
Commonly, polynomials are also used by other numerical methods for
partial differential equations, such as the finite difference method
and the spectral method. Nevertheless, polynomial approximations are
not always effective, especially for problems with rough
coefficients. In the dissertation, a suitable approximation space
for the solution of elliptic problems with rough coefficients has
been found, which is named as generalized L-spline space. Theoretically, I have developed generalized L-spline approximation
spaces, where L is an operator of order m with rough coefficients, have proved the interpolation error estimate, and have also proved that the generalized L-spline space is an optimal approximation space for the problem L*Lu=f with certain operator L, by using n-widths as the criteria. Numerically, two problems have been tested and the relevant error estimate results are consistent with the shown theoretical results.
Meshless methods are newly developed numerical methods for solving
partial differential equations. These methods partially eliminate
the need of meshing. Meshless methods are considered to have great
potential. However, the need for effective quadrature schemes is a
major issue concerning meshless methods. In our recently published
paper, we consider the approximation of the Neumann problem by
meshless methods, and show that the approximation is inaccurate if
nothing special (beyond accuracy) is assumed about the numerical
integration. We then identify a condition - referred to as the zero
row sum condition. This, together with accuracy, ensure the
quadrature error is small. The row sum condition can be achieved by
changing the diagonal elements of the stiffness matrix. Under row
sum condition we derive an energy norm error estimate for the
numerical solution with quadrature. In the dissertation, meshless
methods are discussed and quadrature issue is explained. Two
numerical experiments are presented in details. Both theoretical and
numerical results indicate that the error has two components; one
due to the meshless methods approximation and the other due to
quadrature
Opial type inequalities for conformable fractional derivative and integral of two functions
In this paper, we establish the Opial type inequalities for conformable fractional derivative and integral of two function and give some results in special cases of alpha
Lyapunov-type inequalities for higher order difference equations with anti-periodic boundary conditions
In this paper, some new Lyapunov-type inequalities for higher order difference equations with anti-periodic boundary conditions are established. The obtained results are used to obtain the lower bounds for the eigenvalues of corresponding equations
Lyapunov-type inequalities for fractional differential equations
The principal aim of this paper is to discuss Lyapunov-type inequalities for fractional differential equations with fractional boundary conditions. Some new Lyapunov's inequalities are established, which almost generalize and improve some earlier results in the literature
Oscillation of second order self-conjugate differential equation with impulses
AbstractIn this paper, we investigate the oscillation of second-order self-conjugate differential equation with impulses(1)(a(t)(x(t)+p(t)x(t-τ))′)′+q(t)x(t-σ)=0,t≠tk,t⩾t0,(2)x(tk+)=(1+bk)x(tk),k=1,2,…,(3)x′(tk+)=(1+bk)x′(tk),k=1,2,…,where a,p,q are continuous functions in [t0,+∞), q(t)⩾0, a(t)>0, ∫t0∞(1/a(s))ds=∞, τ>0, σ>0, bk>-1, 0<t0<t1 <t2<⋯<tk<⋯ and limk→∞tk=∞. We get some sufficient conditions for the oscillation of solutions of Eqs. (1)–(3)
Oscillatory criteria for Third-Order difference equation with impulses
AbstractIn this paper, we investigate the oscillation of Third-order difference equation with impulses. Some sufficient conditions for the oscillatory behavior of the solutions of Third-order impulsive difference equations are obtained
Asymptotic behavior of second-order impulsive differential equations
In this article, we study the asymptotic behavior of all solutions of 2-th order nonlinear delay differential equation with impulses. Our main tools are impulsive differential inequalities and the Riccati transformation. We illustrate the results by an example
Existence of solutions to n-th order neutral dynamic equations on time scales
In this article, we study n-th order neutral nonlinear dynamic equation on time scales. We obtain sufficient conditions for the existence of non-oscillatory solutions by using fixed point theory