This compact set of notes present some basic, abstract results with two examples of boundary-value problems on weak convergence methods for the study of existence of minimizers of essentially convex functionals. The method described here is called the direct method in the calculus of variations. The main results are stated in their simplest possible setting. But, they have many variations that ...

in this work‎, ‎an iterative method based on a matrix form of lsqr algorithm is constructed for solving the linear operator equation $mathcal{a}(x)=b$‎ ‎and the minimum frobenius norm residual problem $||mathcal{a}(x)-b||_f$‎ ‎where $xin mathcal{s}:={xin textsf{r}^{ntimes n}~|~x=mathcal{g}(x)}$‎, ‎$mathcal{f}$ is the linear operator from $textsf{r}^{ntimes n}$ onto $textsf{r}^{rtimes s}$‎, ‎$ma...

The Kronig–Penney model serves to illustrate the formation of energy bands in a periodic solid and appears as a pedagogical example in many textbooks in elementary solid state physics. The model is generally solved either by matching the boundary conditions for the wave functions at the cell boundaries, by a plane-wave expansion of the wave function in the reciprocal lattice space, or even by t...

Euler's one step method is undoubtedly the simplest method for approximating the solution to an ordinary differential equation. It goes something like this: Given a first order initial value problem y = f (x, y), x ∈ (a, b) y(a) = y a we observe that defining x j = a + h(j − 1), for j = 1, 2, · · · , (N + 1) where h = (b − a)/N we have y(x k+1) − y(x k) h ≈ f (x k , y(x k)). Therefore we can wr...

In the present thesis we implement a simple numerical method to solve a onedimensional time-independent Schrödinger equation in any arbitrary periodic potential. We apply this method to study the energy band structure, the Bloch wavefunctions and the 1D-density of states of a simple 1D-superlattice model. The results obtained for this model exhibit nontrivial features of real hetero-crystalline...

Abstract. The simplest pair of spaces P1/P0 is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is enhance the finite element spaces inside a PetrovGalerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and ...

A geometric approach to Sundman transformation defined by basic functions for systems of second-order differential equations is developed and the necessity a change tangent structure means function defining shown. Among other applications such theory we study linearisability system in particular simplest case equation. The illustrated with several examples.