We propose novel algorithms to calculate the inverses of ill-conditioned matrices, which have broad engineering applications. The vector-form of the conjugate gradient method (CGM) is recast into a matrix-form, which is named as the matrix conjugate gradient method (MCGM). The MCGM is better than the CGM for finding the inverses of matrices. To treat the problems of inverting illconditioned mat...

We apply the I-method to prove that the Cauchy problem of a higher-order Schrödinger equation is globally well-posed in the Sobolev space Hs(R) with s > 6/7.

We consider the hyperbolic Yang-Mills equation on the Minkowski space R. Our main result asserts that this problem is globally well-posed for all initial data whose energy is sufficiently small. This solves a longstanding open problem.

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H(R) for −3/10 < s.

in this paper, we investigate the application of the homotopy perturbation method (hpm) for solving a one-dimensional nonlinear inverse heat conduction problem. in this problem the thermal conductivity term is a linear function with respect to unknown heat temperature in bounded interval. furthermore, the temperature histories are unknown at the end point of the interval. this problem is ill-po...

where α is a real constant with 2α/3 ̸∈ Z and T > 0. In (1), all the parameters are normalized except for α. Equation (1) appears as a mathematical model for nonlinear pulse propagation phenomena in various fields of physics, especially in nonlinear optics (see [54], [27] and [1]). So far, equation (1) without the third order derivative, that is, the cubic NLS equation has attracted much mathema...

We study well-posed perturbations of formally ill-posed diffusion equations which are used in image processing, such as the Perona–Malik equation. Our perturbation technique is to consider the diffusion equations as L gradient flows on integral functionals and then modify the inner product from L to a Sobolev inner product. We show that the functional differential equations obtained in this way...