By using the degree theory and the τ-topology of Kryszewski and Szulkin, we establish a version o... more By using the degree theory and the τ-topology of Kryszewski and Szulkin, we establish a version of the Fountain Theorem for strongly indefinite functionals. The abstract result will be applied for studying the existence of infinitely many solutions of two strongly indefinite semilinear problems including the semilinear Schrödinger equation.
We establish the existence of nontrivial solutions to sys- tems of singular Poisson equations in ... more We establish the existence of nontrivial solutions to sys- tems of singular Poisson equations in unbounded domains, under some invariance conditions and singular subcritical growth. The proofs rely on a concentration-compactness argument and on a generalized linking theorem due to Krysewski and Szulkin.
An unconditionally stable alternating direction explicit scheme (ADE) to solve the one-dimensiona... more An unconditionally stable alternating direction explicit scheme (ADE) to solve the one-dimensional unsteady convection-diffusion equation was developed by J. Xie, Z. Lin and J. Zhou in [6]. Aside from being explicit and unconditionally stable, the method is straightforward to implement. In this paper we show extensions of this scheme to higher-dimensions of the convection-diffusion equation subject to Dirichlet boundary conditions. By expressing the equation with a local series expansion over a rectangular grid, a linear system of symbolic equations is obtained which is tedious to solve for manually and we addressed this challenge using symbolic computation. The solutions obtained are explicit closed-form formulas which are then used to iteratively solve the unsteady convectiondiffusion equation by traversing the discrete grid in an alternating direction fashion. Finally, extensions to higher dimensions can be easily deduced from the 2D formulas. We conclude the paper with numerical...
A function and its first two derivatives are estimated by convolutions with well-chosen non-diffe... more A function and its first two derivatives are estimated by convolutions with well-chosen non-differentiable kernels. The convolutions are in turn approximated by Newton–Cotes integration techniques with the aid of a polynomial interpolation based on an arbitrary finite set of points. Precise numerical results are obtained with far fewer points than that in classic SPH, and error bounds are derived.
By using the degree theory and the τ-topology of Kryszewski and Szulkin, we establish a version o... more By using the degree theory and the τ-topology of Kryszewski and Szulkin, we establish a version of the Fountain Theorem for strongly indefinite functionals. The abstract result will be applied for studying the existence of infinitely many solutions of two strongly indefinite semilinear problems including the semilinear Schrödinger equation.
We establish the existence of nontrivial solutions to sys- tems of singular Poisson equations in ... more We establish the existence of nontrivial solutions to sys- tems of singular Poisson equations in unbounded domains, under some invariance conditions and singular subcritical growth. The proofs rely on a concentration-compactness argument and on a generalized linking theorem due to Krysewski and Szulkin.
An unconditionally stable alternating direction explicit scheme (ADE) to solve the one-dimensiona... more An unconditionally stable alternating direction explicit scheme (ADE) to solve the one-dimensional unsteady convection-diffusion equation was developed by J. Xie, Z. Lin and J. Zhou in [6]. Aside from being explicit and unconditionally stable, the method is straightforward to implement. In this paper we show extensions of this scheme to higher-dimensions of the convection-diffusion equation subject to Dirichlet boundary conditions. By expressing the equation with a local series expansion over a rectangular grid, a linear system of symbolic equations is obtained which is tedious to solve for manually and we addressed this challenge using symbolic computation. The solutions obtained are explicit closed-form formulas which are then used to iteratively solve the unsteady convectiondiffusion equation by traversing the discrete grid in an alternating direction fashion. Finally, extensions to higher dimensions can be easily deduced from the 2D formulas. We conclude the paper with numerical...
A function and its first two derivatives are estimated by convolutions with well-chosen non-diffe... more A function and its first two derivatives are estimated by convolutions with well-chosen non-differentiable kernels. The convolutions are in turn approximated by Newton–Cotes integration techniques with the aid of a polynomial interpolation based on an arbitrary finite set of points. Precise numerical results are obtained with far fewer points than that in classic SPH, and error bounds are derived.
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