Solutions Of Linear Equations Research Articles

Existence of Solutions of Linear Set-Valued Integral Equations and Their Properties

Existence of Solutions of Linear Set-Valued Integral Equations and Their Properties

The periodic solution of a second order linear equation

The periodic solution of a second order linear equation

Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-Differential Equations

Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-Differential Equations

EXtended Hybridizable Discontinuous Galerkin (X-HDG) method for linear convection-diffusion equations on unfitted domains

In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method are combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e., the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order (p≤4) XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal (p+1) and super (p+2) convergence properties of HDG while removing the fitting mesh restriction.

On the solution of linear functional two-term equations with shift

This work is dedicated to the study of linear functional equations with shift in Hölder space. Previously, for such operators, conditions for invertibility were found and the inverse operator was constructed by the authors. The operators are used in modeling systems with renewable resources. Here we propose another approach to solving functional equations with shift. With the help of an algorithm, the initial equation is reduced to the first iterated equations, then to the second iterated equation. Continuing this process, we obtain the n-th iterated equation and the limit iterated equation. We prove the theorem on the equivalence of the original and the iterated equations. Based on the analysis of the solvability of the limit equation, we find a solution to the original equation. The solution is the sum of an infinite product and a functional series. The results, and the methods for obtaining them, are transparent and not as cumbersome compared to previous works.

Meromorphic solutions of linear q-difference equations

In this article, we construct explicit meromorphic solutions of first order linear q-difference equations in the complex domain and we describe the location of all their zeros and poles. The homogeneous case leans on the study of four fundamental equations, providing the previous informations in the framework of entire or meromorphic coefficients. The inhomogeneous situation, which stems from the homogeneous one and two fundamental equations, is also described in detail. We also address the case of higher-order linear q-difference equations, using a classical factorization argument. All these results are illustrated by several examples.

Reused LU factorization as a preconditioner for efficient solution of the parabolized stability equations

The parabolized stability equations (PSE) are a widely used, efficient method to calculate the evolution of streamwise traveling instabilities in spatially developing, weakly nonparallel flows. Although the PSE are very economical, computational time can be significant due to the repeated solution of linear systems of equations in each marching step. The linear systems of equations are typically solved by LU factorization because of the moderate size and the sparsity of the matrices. In this paper, instead of repeated calculation of the LU factorization, a single LU factorization is calculated in each marching step and used as a preconditioner for an iterative solver. Numerical experiments are conducted on the plane-marching (3D) incompressible PSE, nonlinear PSE (NLPSE) with explicit discretization of the nonlinear terms, and NLPSE with implicit treatment of the nonlinearities. It is shown that the time spent on the solution of the linear systems of equations was reduced by a factor of 2.5, 4.5–7.5, and 20–60 for the three cases, respectively. The only requirement of the proposed numerical techniques is that the solution varies slowly in the marching direction; therefore it is expected to be applicable to similar boundary layer stability problems or slowly varying parabolic partial differential equations. This is supported by the fact that the numerical method was applied to the Boundary Region Equations (BRE) and the line-marching (2D) PSE, the latter being a much smaller problem than its plane-marching counterpart. In these cases, a factor of 50% and 15–30% decrease in the net time of the linear equation solution is reported, respectively.

Results on the growth of solutions of complex linear differential equations with meromorphic coefficients

The purpose of this paper is the study of the growth of solutions of higher order linear differential equations \(f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left(z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right)f^{\left( k-1\right) }+\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right) }+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=0\) and \(f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left(z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right)f^{\left( k-1\right) }+\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right)}+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=F\left( z\right),\) where \(A_{j,i}\left( z\right) \left( \not\equiv 0\right) \left(j=0,...,k-1;i=1,2\right) ,\) \(F\left( z\right) \) are meromorphic functions of finite order and \(P_{j}\left( z\right) ,Q_{j}\left( z\right) \) \((j=0,1,...,k-1;i=1,2)\) are polynomials with degree \(n\geq 1\). Under some others conditions, we extend the previous results due to Hamani and Belaïdi [1].

Improved fuzzy sparse multi-class least squares support vector machine

The classification problem is a key area of research in machine learning. The Least Squares Support Vector Machine (LSSVM) is an important classifier that is commonly used to solve classification problems. Its widespread use stems from its replacement of the inequality constraint in the Support Vector Machine (SVM) with the equality constraint, which transforms the convex quadratic programming (QP) problem of SVM into the solution of linear equations. However, when dealing with multi-class classification problems, LSSVM faces the challenges of lack of sparsity and sample noises, which can negatively impact its performance. Based on the modeling characteristics and data distribution of the multi-class LSSVM model, this paper proposes two improvements and establishes an improved fuzzy sparse multi-class least squares support vector machine (IF-S-M-LSSVM). The first improvement adopts a non-iterative sparse algorithm, which can delete training sample points to different degrees by adjusting the sparse ratio. The second improvement addresses the impact of sample noise on determining the optimal hyperplane by adding a fuzzy membership degree based on sample density. The advantages of the new model, in terms of training speed and classification accuracy, are verified through UCI machine learning standard data set experiments. Finally, the statistical significance of the IF-S-M-LSSVM model is tested using the Friedman and Bonferroni-Dunn tests.

Solutions of fuzzy advection-diffusion and heat equations by natural adomian decomposition method

In this article, we present an algorithm for computing analytical solutions of linear fuzzy advection-diffusion equations and one-dimensional fuzzy heat equations involving an external source. The fuzzy problems can be solved by using the natural transform and Adomian decomposition method. The results obtained through the natural Adomian decomposition method are calculated in a series form that converges rapidly to the exact solution. To enhance the practicality of our work, we provide examples to illustrate our findings.

Infinite Order of The Solutions of Higher Order Homogeneous Linear Differential Equations with Entire Coefficients Having Completely Regular Growth Property and Bounded Components Fatou Set

The homogeneous higher order complex linear differential equations (n-thCLDEs) with entire functions is considered in this paper. We investigated some conditions that can be put on the coefficients which guarantee that any nonzero solution of such equations has infinite order. The conditions we stated are the completely regular growth (CRG), the characteristic function of some coefficients is approximately equals to the logarithm of its maximum modulus and the Denjoy’s conjecture (DC) property.

Positivity of oscillatory integrals and Hankel transforms

ABSTRACT In consideration of the integral transform whose kernel arises as an oscillatory solution of certain second-order linear differential equation, its positivity is investigated on the basis of Sturm's theory. As applications, positivity criteria are obtained for Hankel transforms as well as trigonometric integrals defined on the positive real line.

Bridging machine learning and weighted residual methods for delay differential equations of fractional order

In this research, an attempt has been made to build a bridge between a machine learning model called Least-Squares Support Vector Regression (LS-SVR) and Weighted Residual Methods (WRMs) to solve delay differential equations in fractional derivative order. Our machine learning algorithm, which is influenced by the collocation method, uses Legendre polynomials as a kernel function to approximate the solution of linear and nonlinear delay differential equations, including the Lane–Emden pantograph equation and some fractional cases. In the proposed algorithm, the estimated solution of delay differential equations is approximated as a linear combination of M degrees of Legendre polynomials and a set of weights that are learned during the fitting process. The roots of the Legendre functions are utilized as training data to develop the algorithm. The effectiveness of our method is demonstrated through the use of numerical graphs and tables. A comparative analysis of the numerical results has shown that the proposed method yields better accuracy and convergence compared to other numerical techniques.

On periodic solutions of linear inhomogeneous differential equations with a small perturbation at the derivative

In a Banach space, using branching theory methods, a periodic solution of a linear inhomogeneous differential equation with a small perturbation at the derivative (perturbed equation) is constructed. Under the condition of presence of a complete generalized Jordan set, the uniqueness of this periodic solution is proven. It is shown that when a small parameter is equal to zero and certain conditions are met, the periodic solution of the perturbed equation transforms into the family of periodic solutions of the unperturbed equation. The result is obtained by representing the perturbed equation as an operator equation in Banach space and applying the theory of generalized Jordan sets and modified Lyapunov-Schmidt method. As is known, the latter method reduces the original problem to study of the Lyapunov-Schmidt resolving system in the root subspace. In this case, the resolving system splits into two inhomogeneous systems of linear algebraic equations, that have unique solutions at ε≠0 , and 2n -parameter families of real solutions at ε=0 , respectively.

NUMERICAL ACCURACY OF FREDHOLM LINEAR INTEGRO-DIFFERENTIAL EQUATIONS BY USING ADOMIAN DECOMPOSITION METHOD, MODIFIED ADOMIAN DECOMPOSITION METHOD AND VARIATIONAL ITERATION METHOD

In this article, we present as a comparative result of Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM) and Variational Iteration Method (VIM). These methods used for developed to find the analytical approximate solution of linear Fredholm integro-differential equations. The main purpose of this paper was to show a better method for Numerical equations which does not give easily analytical solution. So, in this paper, we find approximate solutions of linear Fredholm integro-differential equations. We explain the convergence of ADM, MADM and VIM by using examples of a deterministic model by graphs and tables. All the calculations performed by the help of MATLAB (2018) Version 9.4.

Multi-phase iterative learning control for high-order systems with arbitrary initial shifts

Aiming at the second-order tracking system with arbitrary initial shifts, this paper presents a multi-phase iterative learning control strategy. Firstly, utilizing the form of solution of the second-order non-homogeneous linear differential equation with constant coefficients and the initial shifts, we can select the appropriate control gain to ensure that the second-order systems are stable and reach the stable output after a fixed time. Secondly, on the premise that the second-order systems have reached the fixed output, two methods are proposed for rectifying the fixed shift, namely, shifts rectifying control and varied trajectory control. Theoretical analysis shows that the multi-stage iterative learning control strategy proposed in this paper can ensure that the second-order systems achieve complete tracking in the specified interval. Finally, the simulation examples affirm the validation of the designed algorithms.

Algebraic Solution of Tropical Best Approximation Problems

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

An optimal control problem for resource utilisation by microorganisms

Decomposition of organic matter controls the flow of carbon and nutrients in terrestrial and aquatic ecosystems. Several kinetic laws have been proposed to describe decomposition rates, but they neglect adaptation of the microbial decomposer to environmental conditions. Here we formalise decomposition as an optimal control problem by assuming that microorganisms regulate the uptake rate of a substrate to maximise their growth over the period of decomposition. The result is an optimal control problem consisting of two differential equations and auxiliary conditions that determine the optimal value of the control variable (the uptake rate), the remaining substrate at any given time, and the optimal completion time. This problem serves as a case study to illustrate the solution of differential equations and optimal control problems for students in undergraduate courses. The mathematical analysis of the problem requires rewriting the differential equations in reverse time along with the solution of a nonhomogeneous linear first order differential equation. We then return to modelling with some biologically motivated questions about how the parameters of the model representing environmental conditions and microbial functional traits affect the outcome. Finally, we discuss alternative ways to use the material with students.

New semi-analytical approach and its convergence analysis for a classical hyperbolic fragmentation model: A homotopy perturbation method

In the past few decades, the homotopy perturbation method (HPM) has been used intensively for finding analytical and semi-analytical solutions for linear and nonlinear differential equations. This method is well-known for its flexibility and efficiency to solve complicated physical problems. Moreover, HPM offers physical insight into the underlying dynamics of the system, as well as on the evolution of solutions. In this work, HPM is employed to obtain semi-analytic solutions for the hyperbolic particle-fragmentation equation. A recursive scheme is derived for finding the approximate solution. Mathematical formulation of the scheme is further studied by conducting a detailed convergence analysis of the method. Different physics-embedded test problems are solved using this recursive scheme and their series solutions are obtained. The proposed method is validated by comparing the approximated solutions with the exact results. For two test problems, new series solutions are presented in closed form. New analytical solutions are obtained for the particle number density c(x,t), with breakage kernels ℱ(x,y)=1,x+y, and initial condition c(0,x)=exp(−x)x. Moreover, the analytical solutions for ℱ(x,y)=1x+y corresponding to initial conditions c(0,x)=exp(−x) and exp(−x)x are derived for the first time in literature. Due to the non-availability of the analytical solutions of the particle density functions, the results are verified against the finite volume scheme (Saha and Bück, 2021). It is observed that the proposed method predicts the particle density as well as significant integral moments with high accuracy by considering only a fewer number of terms in the series solution.

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.