• 대한수학회보
• /
• 제56권2호
• /
• pp.461-470
• /
• 2019

In this note, we consider the growth of solutions to second order and higher order linear complex differential equations on annuli instead of the complex plane. We establish several theorems that are analogues of the results in the complex plane.

• Journal of applied mathematics & informatics
• /
• 제41권1호
• /
• pp.167-179
• /
• 2023

In this paper, we investigate solutions of equations which are linear (p, q)-difference equation of higher order by using (p, q)-derivative and integral. We also derive the solution of equation in the case of second order.

• Computers and Concrete
• /
• 제7권4호
• /
• pp.317-329
• /
• 2010

The paper describes localization of deformation in a bar under tensile loading. The material of the bar is considered as non-linear viscous elastic and the bar consists of two symmetric halves. It is assumed that the model represents behavior of the quasi-brittle viscous material under uniaxial tension with different loading rates. Besides that, the bar could represent uniaxial stress-strain law on a single plane of a microplane material model. Non-linear material property is taken from the microplane material model and it is coupled with the viscous damper producing non-linear Maxwell material model. Mathematically, the problem is described with a system of two partial differential equations with a non-linear algebraic constraint. In order to obtain solution, the system of differential algebraic equations is transformed into a system of three partial differential equations. System is subjected to loadings of different rate and it is shown that localization occurs only for high loading rates. Mathematically, in such a case two solutions are possible: one without the localization (unstable) and one with the localization (stable one). Furthermore, mass is added to the bar and in that case the problem is described with a system of four differential equations. It is demonstrated that for high enough loading rates, it is the added mass that dominates the response, in contrast to the viscous and elastic material parameters that dominated in the case without mass. This is demonstrated by several numerical examples.

• Journal of applied mathematics & informatics
• /
• 제13권1_2호
• /
• pp.53-65
• /
• 2003

Here we present a study of small-amplitude, shallow water waves on sloping beds. The beds considered in this analysis are linear and quadratic in nature. First we start with stating the relevant governing equations and boundary conditions for the theory of water waves. Once the complete prescription of the water-wave problem is available based on some assumptions (like inviscid, irrotational flow), we normalize it by introducing a suitable set of non-dimensional variables and then we scale the variables with respect to the amplitude parameter. This helps us to characterize the various types of approximation. In the process, a summary of equations that represent different approximations of the water-wave problem is stated. All the relevant equations are presented in rectangular Cartesian coordinates. Then we derive the equations and boundary conditions for small-amplitude and shallow water waves. Two specific types of bed are considered for our calculations. One is a bed with constant slope and the other bed has a quadratic form of surface. These are solved by using separation of variables method.

• KSII Transactions on Internet and Information Systems (TIIS)
• /
• 제15권9호
• /
• pp.3458-3481
• /
• 2021

Existing methods for blind identification of linear block codes without a candidate set are mainly built on the Gauss elimination process. However, the fault tolerance will fall short when the intercepted bit error rate (BER) is too high. To address this issue, we apply the reverse algebra approach and propose a novel "two-step-screening" algorithm by solving the linear error equations on the binary Galois field, or GF(2). In the first step, a recursive matrix partition is implemented to solve the system linear error equations where the coefficient matrix is constructed by the full codewords which come from the intercepted noisy bitstream. This process is repeated to derive all those possible parity-checks. In the second step, a check matrix constructed by the intercepted codewords is applied to find the correct parity-checks out of all possible parity-checks solutions. This novel "two-step-screening" algorithm can be used in different codes like Hamming codes, BCH codes, LDPC codes, and quasi-cyclic LDPC codes. The simulation results have shown that it can highly improve the fault tolerance ability compared to the existing Gauss elimination process-based algorithms.

• Kyungpook Mathematical Journal
• /
• 제28권2호
• /
• pp.185-196
• /
• 1988

We study existence of polynomial integrating factors and solutions F(x, y)=c of first order nonlinear differential equations. We characterize the homogeneous case, and give algorithms for finding existence of and a basis for polynomial solutions of linear difference and differential equations and rational solutions or linear differential equations with polynomial coefficients. We relate singularities to nature of the solution. Solution of differential equations in closed form to some degree might be called more an art than a science: The investigator can try a number of methods and for a number of classes of equations these methods always work. In particular integrating factors are tricky to find. An analogous but simpler situation exists for integrating inclosed form, where for instance there exists a criterion for when an exponential integral can be found in closed form. In this paper we make a beginning in several directions on these problems, for 2 variable ordinary differential equations. The case of exact differentials reduces immediately to quadrature. The next step is perhaps that of a polynomial integrating factor, our main study. Here we are able to provide necessary conditions based on related homogeneous equations which probably suffice to decide existence in most cases. As part of our investigations we provide complete algorithms for existence of and finding a basis for polynomial solutions of linear differential and difference equations with polynomial coefficients, also rational solutions for such differential equations. Our goal would be a method for decidability of whether any differential equation Mdx+Mdy=0 with polynomial M, N has algebraic solutions(or an undecidability proof). We reduce the question of all solutions algebraic to singularities but have not yet found a definite procedure to find their type. We begin with general results on the set of all polynomial solutions and integrating factors. Consider a differential equation Mdx+Ndy where M, N are nonreal polynomials in x, y with no common factor. When does there exist an integrating factor u which is (i) polynomial (ii) rational? In case (i) the solution F(x, y)=c will be a polynomial. We assume all functions here are complex analytic polynomial in some open set.

• 제어로봇시스템학회:학술대회논문집
• /
• 제어로봇시스템학회 1993년도 한국자동제어학술회의논문집(국제학술편); Seoul National University, Seoul; 20-22 Oct. 1993
• /
• pp.480-485
• /
• 1993

A method for obtaining optimal orbital maneuvers of a space vehicle has been developed by combining feedback linearization method with the elegance of the Lambert's theorem. To obtain solutions to nonlinear orbital maneuver problems. The full nonlinear equations of motion for space vehicle in polar coordinate system are transformed exactly into a controllable linear set in Brunovsky canonical form by using feedback linearization by choosing position vector as fully observable output vector. These equations are used to pose a linear optimal tracking problem with a solutions to Lambert's problem and a linear analytical solution of continuous low thrust problem as reference trajectories.

• Journal of applied mathematics & informatics
• /
• 제9권2호
• /
• pp.543-559
• /
• 2002

In this paper we formulate the linear theory for compressible fluids in cylindrical geometry with small perturbation at the material interface. We derive the first order equations in the smooth regions, boundary conditions at the shock fronts and the contact interface by linearizing the Euler equations and Rankine-Hugoniot conditions. The small amplitude solution formulated in this paper will be important for calibration of results from full numerical simulation of compressible fluids in cylindrical geometry.

• 대한수학회논문집
• /
• 제35권1호
• /
• pp.229-241
• /
• 2020

For a second order linear differential equation f" + A(z)f' + B(z)f = 0, with A(z) and B(z) being transcendental entire functions under some restrictions, we have established that all non-trivial solutions are of infinite order. In addition, we have proved that these solutions, with a condition, have exponent of convergence of zeros equal to infinity. Also, we have extended these results to higher order linear differential equations.

• Journal of applied mathematics & informatics
• /
• 제35권3_4호
• /
• pp.399-411
• /
• 2017

We first provide the linear operator equations corresponding to the Tikhonov regularization image denoising problems with different regularization terms, and then we propose how to choose Kronecker product preconditioners which are required for accelerating the $G{\ell}$-PCG method. Next, we provide how to apply the $G{\ell}$-PCG method with Kronecker product preconditioner to the linear operator equations. Lastly, we provide numerical experiments for image denoisng problems to evaluate the effectiveness of the $G{\ell}$-PCG with Kronecker product preconditioner.