Exponential Fitted Methods for the Numerical Solution of the Schrödinger Equation

A new sixth-order Runge-Kutta type method is developed for the numerical integration of the radial Schrödinger equation and of the coupled differential equations of the Schrödinger type. The formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions. We give a comparative error analysis with other sixth order exponentially fitted methods. The theoretical and numerical results indicate that the new method is more accurate than the other exponentially fitted methods.     

The Continued Fraction Solution of the Matrix Equation AX-XB=C

Let K[λ] denote the polynomial ring over the number field K.Suppose thatf(λ)and g(λ)are coprime in K[λ] and deg f(λ)≥1.By the Euclidean algorithmwe obtain the continued fraction     

On a Solution of the Quaternion Matrix Equation X - A X B = C and Its Application

This paper first studies the solution of a complex matrix equation X - AXB = C, obtains an explicit solution of the equation by means of characteristic polynomial, and then studies the quaternion matrix equation X - A X B = C, characterizes the existence of a solution to the matrix equation, and derives closed-form solutions of the matrix equation in explicit forms by means of real representations of quaternion matrices. This paper also gives an application to the complex matrix equation X - AXB =C.     

On the Solution of the Operator Equation AT-TB=S

Let H, K be two Hilbert spaces over complex field A. Let B(HI, K) denote the set of all bounded linear operators from H to K. If H=K, we write B(H) instead of B(H, K). Consider the operator defined by the equation     

Numerical Solution of the Lippmann–Schwinger Equation by Approximate Approximations

A new method for the numerical solution of volume integral equations is proposed and applied to a Lippmann–Schwinger type equation in diffraction theory. The approximate solution is represented as a linear combination of the scaled and shifted Gaussian. We prove spectral convergence of the method up to some negligible saturation error. The theoretical results are confirmed by a numerical experiment.     

On The Maximal-Like Solution of Matrix Equation X ＋A^＊X^-2A=I^＊

In this paper,we study several iterative methods for finding the maximal-like solution of the matrix equation X A~*X~(-2)A=I,and deduce some properties of the maximal-like solution with these methods.     

A Remark on the Self-similar Solution of Semi-linear Schrodinger Equation

In this paper we consider the Cauchy problem to the following semi-linear Schrdinger equation     

The Least Square Solutions to the Quaternion Matrix Equation AX=B

61.IntroductionTheconsiderableprogresseshavebeenmadeinthetheoryofmatricesoverskewfields-However,sofar,wehavenotseenanyofstudyresultsofthe1eastsquareproblemofmatricesoverskewfields.Thematrixequation'AX=B,(l)whichisveryimportant,wereinvestigateddeeplyin[1~3J.lnthispaper,wedefineanormofarealquaternionmatrix,giveexpressionsoftheleastsquaresolutionsofthequaternionma-trixequation(l)andtheequationwiththeconstraintconditionDX=E.Throughoutthispaper,wedenotetherealquaternionfieldbyH,thesetofallmXnma…     

Uniqueness of a Probability Solution to the Kolmogorov Equation with a Diffusion Matrix Satisfying Dini’s Condition

Doklady Mathematics - Abstract—In this note we study the stationary Kolmogorov equation and prove that, in the case where the diffusion matrix satisfies Dini’s condition and the drift...     

Least-Squares Solution with the Minimum-Norm for the Matrix Equation A^TXB ＋ B^TX^TA = D and Its Applications

An efficient method based on the projection theorem,the generalized singular value decompositionand the canonical correlation decomposition is presented to find the least-squares solution with the minimum-norm for the matrix equation A~TXB B~TX~TA=D.Analytical solution to the matrix equation is also derived.Furthermore,we apply this result to determine the least-squares symmetric and sub-antisymmetric solution ofthe matrix equation C~TXC=D with minimum-norm.Finally,some numerical results are reported to supportthe theories established in this paper.     

Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x ²,…,x ^p ,exp?(±wx),xexp?(±wx),…,x ^m exp?(±w x)} and the second is based on the exact integration of the functions of the form: {1,x,x ²,…,x ^p ,exp?(±wx),exp?(±2wx),…,exp?(±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrödinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrödinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrödinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrödinger equation. Finally, numerical results for the resonance problem of the radial Schrödinger equation are presented.     

Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.     

Oscillation of Numerical Solution in the Runge-Kutta Methods for Equation x＇（t） = ax（t） ＋ aox（[t]）

The paper de ls with oscillation of Runge-Kutta methods for equation x＇（t） = ax（t） ＋ aox（[t]）. The conditions of oscillation for the numerical methods are presented by considering the characteristic equation of the corresponding discrete scheme. It is proved that any nodes have the same oscillatory property as the integer nodes. Furthermore, the conditions under which the oscillation of the analytic solution is inherited by the numerical solution are obtained. The relationships between stability and oscillation are considered. Finally, some numerical experiments are given.     

Derivative Estimates for the Solution of Hyp erb olic A?ne Sphere Equation

Considering the hyperbolic affine sphere equation in a smooth strictly convex bounded domain with zero boundary values, the sharp derivative estimates of any order for its convex solution are obtained....     

A Criterion for the Uniqueness of the Solution of the Tricomi Problem for the Multidimensional Lavrent’ev–Bitsadze Equation

Differential Equations - An example of a mixed domain in which the solution of the Tricomi problem is trivial is given. A criterion for the uniqueness of the classical solution of this problem is...