Software for the numerical solution of first-order partial differential equations

Partial differential equations of the first order, arising in applied problems of optics and optoelectronics, often contain coefficients that are not defined by a single analytical expression in the entire considered domain. For example, the eikonal equation contains the refractive index, which is described by various expressions depending on the optical properties of the media that fill the domain under consideration. This type of equations cannot be analysed by standard tools built into modern computer algebra systems, including Maple.The paper deals with the adaptation of the classical Cauchy method of integrating partial differential equations of the first order to the case when the coefficients of the equation are given by various analytical expressions in the subdomains G1, . . . , Gk , into which the considered domain is divided. In this case, it is assumed that these subdomains are specified by inequalities. This integration method is implemented as a Python program using the SymPy library. The characteristics are calculatednumerically using the Runge-Kutta method, but taking into account the change in the expressions for the coefficients of the equation when passing from one subdomain to another. The main functions of the program are described, including those that can be used to illustrate the Cauchy method. The verification was carried out by comparison with the results obtained in the Maple computer algebra system.

A New Method of Integrating the Equations of Autonomous Strapdown INS

We propose the new version of separating the process of integrating the differential equations, which describe the functioning of the strapdown inertial navigation system (SINS) in the normal geographic coordinate system (NGCS), into rapid and slow cycles. In this version, the vector of the relative velocity of an object is represented as a sum of a rapidly changing component and a slowly changing component. The equation for the rapidly changing component of the relative velocity includes the vectors of angular velocities of the Earth’s rotation, NGCS rotation, and, at the same time, the vectors of the apparent acceleration and gravity acceleration, because these accelerations partially balance each other, and at rest relative to the Earth are balanced completely. The equation of the slowly changing component of the relative velocity includes only the vector of angular velocity of the Earth’s rotation and the vector of NGCS rotation. The quaternion orientation of an object relative to the NGCS is represented as a product of two quaternions: a rapidly changing one, which is determined by the absolute angular velocity of an object, and slowly changing one, which is determined by the angular velocity of the NGCS. The right parts of the equations for each group of variables depend on the rapidly changing and slowly changing variables. In order to enable the independent integration of the slow and rapid cycle equations, the algorithm have been developed for integrating the equations using the predictor and corrector for the cases of instantaneous and integral information generated by SINS sensors. At each predictor step the Euler method is used to estimate the longitude, latitude and altitude of an object, slowly changing component of the relative velocity, and slowly changing multiplier of the orientation quaternion at the rightmost point of the slow cycle. Then the Euler-Cauchy method is used to integrate the equations for the rapidly changing components on the rapid cycle intervals, which are present in the slow cycle. The necessary values of the slowly changing components in the intermediate points are calculated using the formulas of linear interpolation. After the rapidly changing components are estimated at the rightmost point of the slow cycle, at the corrector step the Euler-Cauchy method is used to refine the values of the slowly changing components at the rightmost point of the slow cycle. Note that at the beginning of each slow cycle step the slowly changing component of velocity is equal to the value of the relative velocity of an object, and the rapidly changing component is zero. Similarly, at the beginning of each slow cycle step the slowly changing multiplier of object’s orientation quaternion equals to the quaternion of orientation of an object relative to the NGCS, and the rapidly changing multiplier of the orientation of an object has its scalar part equal to one, and its vector part equal to zero (this formula is derived from the quaternion formula for adding the finite rotations). SINS on a stationary base had been simulated in the presence of perturbations for a large time interval for a diving object, which drastically changes its height over short time periods.