Solutions Of Linear Equations Research Articles

SPECTRAL METHODS FOR SOLVING DIFFERENTIAL AND FUNCTIONAL EQUATIONS

The operator approach previously developed for the spectral method using Legendre polynomials is generalized here to any systems of basis functions (not necessarily orthogonal) that satisfy two conditions: the result of the operation of multiplication by x or differentiation with respect to x is expressed in the same functions. All systems of classical orthogonal polynomials meet these conditions. In particular, a spectral method utilizing Chebyshev polynomials is constructed, which is most efficient for numerical calculations. This method is applied for the numerical solution of linear functional equations that arise in generalized series summation problems, aswell as in problems of analytic continuation of discrete mappings. It is also shown how these methods solve nonstandard and nonlinear boundary value problems for which conventional algorithms are not applicable.

Standard solutions of complex linear differential equations

A meromorphic solution of a complex linear differential equation (with meromorphic coefficients) for which the value zero is the only possible finite deficient/deviated value is called a standard solution. Conditions for the existence and the number of standard solutions are discussed for various types of deficient and deviated values.

Growth Estimates of Solutions of Linear Differential Equations with Dominant Coefficient of Lower (α, β, γ)-order

Growth Estimates of Solutions of Linear Differential Equations with Dominant Coefficient of Lower (α, β, γ)-order

Stability and convergence computational analysis of a new semi analytical-numerical method for fractional order linear inhomogeneous integro-partial-differential equations

Abstract The motive behind this research work, is to originate a semi-analytical-numerical approach for the solution of fractional order linear integro partial differential equations (FOLIPDEs), especially in-homogeneous FOLIPDEs of different types, like fractional version of Fredholm and Volterra type IEs etc. To attain the said purpose, we will study some fractional formulations of linear model integral equations from the existing literature. Then we will describe the proposed semi analytical-numerical procedure alongwith its stability analysis and convergence properties. We will then prove with the aid of some particular numerical examples that the said approach is not only lucid and efficient but also precise. The outcomes of the proposed technique will show that this scheme has notable prospectives for solving a large class of FOLIEs. At the end, the contribution of this work in the advancements of semi analytical-numerical approaches for
the solution of fractional order linear integro partial differential equations will be laid down and discussion for future research in this field.

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

On Asymptotic Behavior of Solutions of Linear Multidimensional Stochastic Differential Equations with Multiplicative Noise

Representability of G-functions as rational functions in hypergeometric series

Fresán and Jossen have given a negative answer to a question of Siegel about the representability of every E-function as a polynomial with algebraic coefficients in E-functions of type Fqp[a_;b_;γxq−p+1] with q≥p≥0, γ∈Q‾ and rational parameters a_,b_. In this paper, we study, in a more general context, a similar question for G-functions asked by Fischler and the second author: can every G-function be represented as a polynomial with algebraic coefficients in G-functions of type μ(x)⋅pFp−1[a_;b_;λ(x)] with p≥1, rational parameters a_,b_ and μ,λ algebraic over Q(x) with λ(0)=0? They have shown the answer to be negative under a generalization of Grothendieck's Period Conjecture and a technical assumption on the λ's. Using differential Galois theory, we prove that, for every N∈N, there exists a G-function which can not be represented as a rational function with coefficients in C(x)‾ of solutions of linear differential equations with coefficients in C(x) and at most N singularities in P1(C). As a corollary, we deduce that not all G-functions can be represented as a rational function in hypergeometric series of the above mentioned type, when the λ's are rational functions with degrees of their numerators and denominators bounded by an arbitrarily large fixed constant. This provides an unconditional negative answer to the question asked by Fischler and the second author for such λ's.

Fluctuating hydrodynamics of an autophoretic particle near a permeable interface

We study the autophoretic motion of a spherical active particle interacting chemically and hydrodynamically with its fluctuating environment in the limit of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic fields can be expressed in terms of integrals. The resulting boundary-domain integral equations provide a direct way of obtaining the traction on the particle, requiring the solution of linear integral equations. An exact solution for the chemical and hydrodynamic problems is obtained for a particle in an unbounded domain. For motion near boundaries, we provide corrections to the unbounded solutions in terms of chemical and hydrodynamic Green's functions, preserving the dissipative nature of autophoresis in a viscous fluid for all physical configurations. Using this, we give the fully stochastic update equations for the Brownian trajectory of an autophoretic particle in a complex environment. First, we analyse the Brownian dynamics of particles capable of complex motion in the bulk. We then introduce a chemically permeable planar surface of two immiscible liquids in the vicinity of the particle and provide explicit solutions to the chemo-hydrodynamics of this system. Finally, we study the case of an isotropically phoretic particle hovering above an interface as a function of interfacial solute permeability and viscosity contrast.

Differences in Swedish and Norwegian pre-service teachers’ explanations of solutions of linear equations

Solving linear equations is a cornerstone in the learning of algebra. There are two main strategies for solving a linear equation, ‘swap sides swap signs’ (SSSS) and ‘do the same to both sides’ (DSBS). While SSSS can often be more efficient for solving equations, DSBS has been shown to better promote the learning of algebra. Thus, the preference of SSSS or DSBS might depend on the purpose of solving equations. Since both approaches are common, mathematics teachers, and thus also pre-service teachers (PSTs), must be familiar with both SSSS and DSBS. This study draws on data from 161 Swedish and 146 Norwegian PSTs. They were given a correct but short and unannotated solution to the linear equation x + 5 = 4x − 1. The PSTs were invited to explain the provided solution for a fictive friend. Of the Norwegian PSTs, 2/3 explained the additive steps in the solution by SSSS, while only 1/3 of the Swedish PSTs applied SSSS. Consequently, DSBS was more frequent among the Swedish PSTs regarding the additive steps. However, in the final, multiplicative step, 3/4 of the Norwegian PSTs invoked DSBS. On the contrary, among the Swedish PSTs, the proportion applying DSBS for the multiplicative step decreased, and it was common to provide an incomplete explanation of the final operation. We also analysed how mathematics textbooks for secondary school presented how to solve linear equations. In Sweden, all textbooks utilised DSBS through the whole solution for all years in secondary school. This also applied for Norwegian textbooks for the first two years of lower secondary school. However, in last year of lower secondary school, they changed their approach and promoted an SSSS strategy in additive steps, while DSBS was still suggested for multiplicative steps. This might explain the differences between the two countries regarding the PSTs’ preferences of solution strategies. We suggest that these results can be useful for teacher education, since increased awareness of PSTs’ pre-knowledge is beneficial to support their development of teaching linear equations.

NUMA-aware parallel sparse LU factorization for SPICE-based circuit simulators on ARM multi-core processors

In circuit simulators that resemble the Simulation Program with Integrated Circuit Emphasis (SPICE), one of the most crucial steps is the solution of numerous sparse linear equations generated by frequency domain analysis or time domain analysis. The sparse direct solvers based on lower-upper (LU) factorization are extremely time-consuming, so their performance has become a significant bottleneck. Despite the existence of some parallel sparse direct solvers for circuit simulation problems, they remain challenging to adapt in terms of performance and scalability in the face of rapidly evolving parallel computers with multiple NUMA hardware based on ARM architecture. In this paper, we introduce a parallel sparse direct solver named HLU, which re-examines the performance of the parallel algorithm from the viewpoint of parallelism in pipeline mode and the computing efficiency of each task. To maximize task-level parallelism and further minimize the thread waiting time, HLU devises a fine-grained scheduling method based on an elimination tree in pipeline mode, which employs depth-first search (DFS-like) to iteratively search for parent tasks and then place dependent tasks in the same task queue. HLU also suggests two NUMA node affinity strategies: thread affinity optimization based on NUMA nodes topology to guarantee computational load balancing and data affinity optimization to enable effective memory placement when threads access data. The rationality and effectiveness of the sparse solver HLU are validated by the SuiteSparse Matrix Collection. In comparison with KLU and NICSLU, the experimental results and analysis show that HLU attains a speedup of up to 9.14× and 1.26x (geometric mean) on a Huawei Kunpeng 920 Server, respectively.

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

A novel efficient analytical technique and its application to fractional Cauchy reaction–diffusion equations

In this research paper, we suggest a novel analytical method to obtain the approximate solutions of linear and nonlinear differential equations of arbitrary order. The proposed technique is a good combination of Kharrat–Toma transform and q-homotopy analysis method. Additionally, we also find out the solution of Cauchy reaction–diffusion equation associated with regularized version of Hilfer–Prabhakar fractional derivative. The Cauchy reaction–diffusion equation is broadly used to describe the dynamical processes in physics, geology, biology, ecology, etc. Some characteristic properties of the considered model and existence as well as the uniqueness of the solutions are also discussed. The outcomes of the considered model are exhibited graphically to show the accuracy and efficiency of the suggested method.

Novel results from quadratically nonlinear elastic wave models using Murnaghan’s potential

AbstractIn this article, we study one, two and three-dimensional nonlinear elastic wave equations using quadratically nonlinear Murnaghan potential. We employ two effective methods for obtaining approximate series solutions the Adomian decomposition and the variational iteration method. These methods have the advantage of not requiring any physical parametric assumptions in the problem. Finally, these methods can generate expansion solutions for linear and nonlinear differential equations without perturbation, linearization, or discretization. The results obtained using the adopted methods along various initial and boundary conditions are in excellent agreement with the numerical results on MATLAB, which show the reliability of our methods to these problems. We came to the conclusion that our methods are accurate and simple to use.

Existence and Uniqueness of Homogeneous Linear Equation Solutions in Supertropical Algebra

This research investigates the existence and uniqueness of solutions to homogeneous linear equations in supertropical algebra. We analyze the structure of supertropical matrices to identify the conditions in which nontrivial solutions exist for the system of equations A⊗𝒙 ⊨ 𝜀, where A is a matrix over a supertropical semiring and x is a vector. By applying determinant-based criteria, we demonstrate how tropical and supertropical values influence the solution space. The research applies theorems that determine the presence of trivial and nontrivial solutions and uses examples to illustrate practical methods for solving homogeneous matrix systems. This highlights the distinct characteristics of supertropical algebra compared to classical linear algebra. Our findings provide a deeper insight into solution behaviors in supertropical systems, paving the way for further research in tropical mathematics.

Discrete-time distributed algorithms for solving linear equations via layered coordination

This paper is concerned with the problem of designing discrete-time distributed algorithms for solving large-scale linear equations via layered coordination. A row-column decomposition and a column-row decomposition are provided to partition the augmented matrix associated with the linear equations into several block matrices, respectively. By assigning an equation solver layer to computation and a data integration layer to data exchanges, a row-column decomposition-based discrete-time distributed algorithm and a column-row decomposition-based discrete-time distributed algorithm are proposed for solving large-scale linear equations, respectively. It is shown that the distributed algorithms proposed can reach a consensus exponentially on one of the solutions of linear equations. Finally, the effectiveness of the distributed algorithms proposed is validated via the numerical simulation of the power flow calculation of power systems and the problem of solving the large-scale linear equations.

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Efficient numerical solution of linear Fredholm integro-differential equations via backward finite difference and Nyström methods

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

Numerical solution to the Neumann problem in a Lipschitz domain, based on random walks

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods.

Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the formf(∂t)ϕ=J(t),t≥0, where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the functionϕ(t)=∑n≥1μ(n)nhJ(t−ln⁡(n)), in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equationζ(∂t+h)ϕ=J(t),t≥0, in which ζ is the Riemann zeta function and h>1. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space L2 and the Hardy space H2.