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Consider a general linear Hamiltonian system

This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let In another article to appear, we will prove that when

This monograph is the first to provide a comprehensive, self-contained and rigorous presentation of some of the most powerful preconditioning methods for solving finite element equations in a common block-matrix factorization framework.

Let On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at

We present a randomized algorithm that on input a symmetric, weakly diagonally dominant $n$-by-$n$ matrix $A$ with $m$ nonzero entries and an $n$-vector $b$ produces an ${\\tilde{x}} $ such that $\\|{\\tilde{x}} - A^{\\dagger} {b} \\|_{A} \\leq \\epsilon \\|A^{\\dagger} {b}\\|_{A}$ in expected time $O (m \\log^{c}n \\log (1/\\epsilon))$ for some constant $c$. By applying this algorithm inside the inverse power method, we compute approximate Fiedler vectors in a similar amount of time. The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by Vaidya in 1990. For any symmetric, weakly diagonally dominant matrix $A$ with nonpositive off-diagonal entries and $k \\geq 1$, we construct in time $O (m \\log^{c} n)$ a preconditioner $B$ of $A$ with at most $2 (n - 1) + O ((m/k) \\log^{39} n)$ nonzero off-diagonal entries such that the finite generalized condition number $\\kappa_{f} (A,B)$ is at most $k$, for some other constant $c$. In the special case when the nonzero structure of the matrix is planar the corresponding linear system solver runs in expected time $O (n \\log^{2} n + n \\log n \\ \\log \\log n \\ \\log (1/\\epsilon))$. We hope that our introduction of algorithms of low asymptotic complexity will lead to the development of algorithms that are also fast in practice. [PUBLICATION ABSTRACT]

The objective of this study is to develop an analytical model for 1D gas‐water flow with evaporation of water into the mobile gas phase. The evaporation rate is proportional to the fluid‐fluid interfacial area, which is a function of the water saturation. Introduction of saturation‐dependent potential allows reducing the 2 × 2 non‐linear system of PDEs to one hyperbolic equation. The saturation distribution is obtained by non‐linear method of characteristics. The first integral allows for explicit expression of the vapor concentration versus saturation. It was shown that typical properties of rock‐CO2‐water system, the evaporation time has order of magnitude of millions of pore volumes injected. Close match was observed between a series of three corefloods and the analytical model, and the tuned model coefficients belong to their common intervals. This validates the developed model for 1D gas‐water flow with water evaporation into the injected gas. Plain Language Summary During injection of carbon dioxide or other gases into porous media containing small fractions of water, the water phase will slowly evaporate into the injected gas. This process is important as the gradual drying of the porous media near the injection site can lead to formation damage processes such as fines migration and salt precipitation, resulting in a significant decline in the ability to inject more gas. The evaporation process is dependent on the interfacial area between the gas and water. As the evaporation process progresses, this area gets smaller, and the evaporation rate slows down. In this work, we account for this effect directly, and solve the resulting differential equations, providing simple equations that can be used to predict the rate of water evaporation during gas injection. Key Points Exact solution of gas transport in porous media with non‐equilibrium evaporation kinetics is obtained The new solution relies on introducing a new potential function, which reduces the system of non‐linear PDEs to one equation The resulting exact solution shows good agreement with laboratory data and can predict accurately the evaporation time

This manuscript presents a novel multi-step, tenth-order iterative method for solving fuzzy nonlinear equations, which frequently emerge in a variety of applications such as optimization, decision-making, control theory, and chemical engineering problems. One of the principal challenges in solving these equations lies in the computational demands of computing and inverting the Jacobian matrix at each iteration. The primary advantage of the proposed iterative method is that it obviates the need for Jacobian matrix calculations, thereby markedly reducing the computational complexity associated with solving fuzzy nonlinear Equations. We conduct a thorough convergence analysis and establish that our method achieves tenth-order convergence. The effectiveness and robustness of the developed approach are illustrated through comprehensive numerical examples and real-life application problems, complete with graphical representations. Furthermore, we compare our method with existing tenth-order iterative methods to demonstrate the superior efficiency of our approach.

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.

In this paper, the author considers semilinear elliptic equations of the form $-\\Delta u- \\frac{\\lambda}{|x|^2}u +b(x)\\,h(u)=0$ in $\\Omega\\setminus\\{0\\}$, where $\\lambda$ is a parameter with $-\\infty0$. The author completely classifies the behaviour near zero of all positive solutions of equation (0.1) when $h$ is regularly varying at $\\infty$ with index $q$ greater than $1$ (that is, $\\lim_{t\\to \\infty} h(\\xi t)/h(t)=\\xi^q$ for every $\\xi>0$). In particular, the author's results apply to equation (0.1) with $h(t)=t^q (\\log t)^{\\alpha_1}$ as $t\\to \\infty$ and $b(x)=|x|^\\theta (-\\log |x|)^{\\alpha_2}$ as $|x|\\to 0$, where $\\alpha_1$ and $\\alpha_2$ are any real numbers.