Browsing by Author "Filipov S.M."
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Item A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling(2023-04-01) Filipov S.M.; Hristov J.; Avdzhieva A.; Faragó I.This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower temperature leaves the tank through another pipe. We propose a one-dimensional mathematical model that consists of a nonlinear PDE for the temperature along the solid body, coupled to a linear ODE for the temperature in the tank, the boundary and the initial conditions. All equations are converted into a dimensionless form reducing the input parameters to three dimensionless numbers and a dimensionless function. A steady-state analysis is performed. To solve the transient problem, a nontrivial numerical approach is proposed whereby the differential equations are first discretized in time. This reduces the problem to a sequence of nonlinear two-point boundary value problems (TPBVP) and a sequence of linear algebraic equations coupled to it. We show that knowing the temperature in the system at time level n − 1 allows us to decouple the TPBVP and the corresponding algebraic equation at time level n. Thus, starting from the initial conditions, the equations are decoupled and solved sequentially. The TPBVPs are solved by FDM with the Newtonian method.Item Exceeding Information Targets in Fixed-Form Test Assembly(2021-01-01) Gospodinov I.; Karaibrahimova E.N.; Filipov S.M.This work studies the automated assembly of ability estimation test forms (called tests, for short) drawn from an item bank. The goal of fixed-from test assembly is to generate a large number of different tests with information functions that meet a target information function. Thus, every test has the same ability estimation error. This work proposes a new way of automated test assembly, namely, drawing tests with information functions that exceed the target. This guarantees that every test has an ability estimation error that is less than the error set by the target. The work estimates the number of target exceeding tests as a function of the number of items in the test. It demonstrates that the number of target exceeding tests is far greater than the number of target meeting tests. A Monte Carlo importance sampling algorithm is proposed for target exceeding test assembly.Item Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method(2019-10-01) Filipov S.M.; Gospodinov I.D.; Faragó I.This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O(N 3 ), where N is the number of mesh-points, to only O(N).Item Shooting-projection method for a small object moving under the influence of a force(2021-09-20) Filipov S.M.; Faragó I.; Avdzhieva A.We consider a small object in 3D moving under the influence of a force that may depend explicitly on time, on the position of the object, and on its velocity. The equations of motion of classical mechanics are assumed to hold. If the position of the object is specified at some initial and some final time, obtaining the trajectory of the object requires the solution of a two-point boundary value problem. To solve the problem various numerical technics can be applied. This paper extends the recently proposed shooting-projection method to 3D. We introduce a Lagrangian from which, applying the principle of least action, the projection trajectory is derived. Analysis of the action reveals the meaning of the projection trajectory. Using the shooting-projection method, the considered two-point boundary value problem is solved for the case of a projectile motion in the presence of air resistance and wind.Item Shooting-projection method for two-point boundary value problems(2017-10-01) Filipov S.M.; Gospodinov I.D.; Faragó I.This paper presents a novel shooting method for solving two-point boundary value problems for second order ordinary differential equations. The method works as follows: first, a guess for the initial condition is made and an integration of the differential equation is performed to obtain an initial value problem solution; then, the end value of the solution is used in a simple iteration formula to correct the initial condition; the process is repeated until the second boundary condition is satisfied. The iteration formula is derived utilizing an auxiliary function that satisfies both boundary conditions and minimizes the H1 semi-norm of the difference between itself and the initial value problem solution.