Solvability Of Equations Research Articles

A Glimpse on Integral Solutions of Ternary Uniform Second Degree Equation 𝟓 (𝐱𝟐 + 𝐲𝟐) − 𝟑𝑥𝑦 = 𝟐𝟎𝟎𝟎𝑧𝟐

Objectives: Many integer solutions to ternary homogeneous quadratic given by are determined. Various choices of integer solutions are secure from beginning to end employing linear modifications and used to simplify expressions. Methods: Different transformations are utilized to determine many solutions in integers for the considered polynomial equation. Findings: Five distinct transformations are applied to obtain choices of integral solutions for the considered second-degree equation having three unknowns. Novelty: The equation in the title has been reduced to a solvable ternary quadratic equation by employing suitable transformation. Mathematics Subject Classification: 11D09 Keywords: Quadratic having three unknowns, Homogeneous quadratic, Integer Solutions, Ternary Quadratic Equation, Pair of equations

Numerical analysis of MHD Casson fluid streaming above a stretching surface with energy transfer

In this study, we have numerically inspected the nature of viscous fluid flow in the Casson fluid with heat and mass transmissions over an elongating sheet. A magnetic field is implemented along the normal direction of the flow. The leading partial differential equations are transmuted into solvable ordinary differential equations using some suitable similarity conversions. Graphical and tabular presentations of the skin friction, the local Sherwood number, the Nusselt number are provided.

Exactly solvable time-dependent oscillator family

This paper proposes a method for constructing exactly solvable time-dependent oscillator families based on the invariant operator method and the duality method. An exactly solvable time-dependent oscillator family contains an infinite number of exactly solvable time-dependent oscillators. As a byproduct, we also construct exactly solvable Ermakov-Pinney equation families.

Strong barriers for weighted quasilinear equations

In potential theory, use of barriers is one of the most important techniques. We construct strong barriers for weighted quasilinear elliptic operators. There are two applications: (i) solvability of Poisson-type equations with boundary singular data, and (ii) a geometric version of Hardy inequality. Our construction method can be applied to a general class of divergence form elliptic operators on domains with rough boundary.

Super-efficient exact Hamiltonian Monte Carlo for the von Mises distribution

Markov Chain Monte Carlo algorithms, the method of choice to sample from generic high-dimensional distributions, are rarely used for continuous one-dimensional distributions, for which more effective approaches are usually available (e.g. rejection sampling). In this work we present a counter-example to this conventional wisdom for the von Mises distribution, a maximum-entropy distribution over the circle. We show that Hamiltonian Monte Carlo with Laplacian momentum has exactly solvable equations of motion and, with an appropriate travel time, the Markov chain has negative autocorrelation at odd lags for odd observables and yields a relative effective sample size bigger than one.

A novel method to solve analytically the non-linear Poisson equation in the inversion layer of a MOSFET

Abstract Despite the enormous importance that the metal oxide semiconductors (MOS) and the field effect transistors (MOSFETs) have in the actual semiconductor technology, the task of finding the analytical solution of the Poisson equation in the inversion layer was not fully accomplished for more than half a century. It is a well-known fact that due to the non-linear nature of this equation, the attempts to solve it got stuck after the first integration. Nevertheless, experimental and applied researchers found ways to characterize, control, and develop MOSFET devices based on approximate models, and numerical calculations. Here I present a new method to analytically solve the nonlinear Poisson equation, in principle, for any charge distribution in the inversion layer of a MOS, and for the charge density of the original Shockley model, in particular. To this purpose, a physical argument related to the charge and field energies in the transient population-inversion process is introduced and a new nonlinear but solvable second order differential equation is obtained, whose solution also solves the original Poisson equation. The analytical results presented here allow us to derive explicit expressions for the electrical potential distribution and, very importantly, for the charge distribution, the inversion layer width and the effective impurity concentration in the depletion layer. The quantum Hall effect and other physical phenomena were discovered in the inversion layer of a MOS. The Hall effect was explained under the assumption that the charge distribution is a two-dimensional electron gas, we show here however that the 2D limit is reached only at high gate voltages. When applied to MOSFET structures, we obtain new expressions for the drain currents in the inversion and depletion layers, as functions of the impurity concentration and the gate voltage for single and double-gate MOSFETs, and we give an insight for multi-gate MOSFETs.

Hybrid Nanofluid Flow over a Shrinking Rotating Disk: Response Surface Methodology

For efficient heating and cooling applications, minimum wall shear stress and maximum heat transfer rate are desired. The current study optimized the local skin friction coefficient and Nusselt number in Al2O3-Cu/water hybrid nanofluid flow over a permeable shrinking rotating disk. First, the governing equations and boundary conditions are solved numerically using the bvp4c solver in MATLAB. Von Kármán’s transformations are used to reduce the partial differential equations into solvable non-linear ordinary differential equations. The augmentation of the mass transfer parameter is found to reduce the local skin friction coefficient and Nusselt number. Higher values of these physical quantities of interest are observed in the injection case than in the suction case. Meanwhile, the increase in the magnitude of the shrinking parameter improved and reduced the local skin friction coefficient and Nusselt number, respectively. Then, response surface methodology (RSM) is conducted to understand the interactive impacts of the controlling parameters in optimizing the physical quantities of interest. With a desirability of 66%, the local skin friction coefficient and Nusselt number are optimized at 1.528780016 and 0.888353037 when the shrinking parameter (λ) and mass transfer parameter (S) are −0.8 and −0.6, respectively.

On some generalizations of Darbo- and Sadovskii- type fixed point theorems in Fréchet spaces

Integral equations form an essential part of nonlinear analysis. In many situations, when seeking solutions to such equations, we utilize fixed-point theorems. Among the most well-known and useful in this regard are the theorems of Darbo and Sadovskii, which generalize the classical results of Schauder and Tikhonov. These theorems have seen many generalizations. The aim of the paper is to strengthen (by weakening the assumptions) several known Darbo- and Sadovskii-type fixed-point theorems in the so-called power version for Fréchet spaces, which are particularly useful for studying the solvability of equations defined on an unbounded interval. A theorem on the existence of solutions to a nonlinear integral equation, illustrating the application of our main result, is also provided.

Globally solvable time-periodic evolution equations in Gelfand–Shilov classes

AbstractIn this paper we consider a class of evolution operators with coefficients depending on time and space variables $$(t,x) \in {\mathbb {T}}\times {\mathbb {R}}^n$$ ( t , x ) ∈ T × R n , where $${\mathbb {T}}$$ T is the one-dimensional torus, and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand–Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on $${\mathbb {R}}^n$$ R n .

A (2 + 1)-Dimensional Integrable Breaking Soliton Equation and Its Algebro-Geometric Solutions+

A new (2 + 1)-dimensional breaking soliton equation with the help of the nonisospectral Lax pair is presented. It is shown that the compatible solutions of the first two nontrivial equations in the (1 + 1)-dimensional Kaup–Newell soliton hierarchy provide solutions of the new breaking soliton equation. Then, the new breaking soliton equation is decomposed into the systems of solvable ordinary differential equations. Finally, a hyperelliptic Riemann surface and Abel–Jacobi coordinates are introduced to straighten the associated flow, from which the algebro-geometric solutions of the new (2 + 1)-dimensional integrable equation are constructed by means of the Riemann θ functions.

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

Solvability of systems of partial fuzzy relational equations revisited – a short note

Fuzzy inference systems have been widely investigated from different perspectives, including their logical correctness. It is not surprising that the logical correctness led mostly to the questions on the preservation of modus ponens. Indeed, whenever such a system processes an input equivalent to one of the rule antecedents, it is natural to expect the modus ponens to be preserved and the inferred output to be identical to the respective rule consequent. This leads to the related systems of fuzzy relational equations where the antecedent and consequent fuzzy sets are known values, the inference is represented either by the direct product related to the compositional rule of inference or by the Bandler-Kohout subproduct, and the fuzzy relation that represents the fuzzy rule base is the unknown element in the equations. The most important question is whether such systems are solvable, i.e., whether there even exists a fuzzy relation that models the given fuzzy rule base in such a way that the modus ponens is preserved.Partiality allows dealing with partially defined truth values which allows to deal with situations, where we cannot define the truth value for a given predicate. The partial logics have been recently extended to partial fuzzy logics and the partial fuzzy set theory has been developed. Partial fuzzy sets then may have undefined membership degrees for some values from the given universe.This background leads naturally to the problem of solvability of partial fuzzy relational equations, which are equations with partial fuzzy sets in the role of antecedents and consequents and with partial fuzzy relation as the model of the given fuzzy rule base. Such a setting led to a recent publication that uncovers the solvability and even the shape of the solutions. In this contribution, we revisit the problem and consider a specific case that allows partiality only in the input. In other words, antecedents, as well as consequents expressing the knowledge, are fully defined. Thus, the model of the fuzzy rules is one of the standard and fully defined relations. We investigate what happens if the input differs from one of the antecedents only by a few undefined values. This mimics the situation when a description of an observed object misses a few values in its feature vector. We show that under specific conditions, we still may preserve the modified modus ponens, i.e., that the inferred output is identical with the fully defined consequent of the respective rule.

Linear operators and construction of symmetric symbols satisfying sum rules of order p

ABSTRACT We present a new method to construct symmetric symbols corresponding to solvable multiscaling equations using linear operators. First, we define a linear operator corresponding to a symbol. We demonstrate that the existence of a solution to a multiscaling equation depends on the 1-eigenspace of the corresponding linear operator. Furthermore, we show how to construct symmetric symbols satisfying the sum rules of orders 1 and 2 by appropriately defining the linear operator on a basis. We also construct symmetric symbols satisfying sum rules of order 3 with degree 3. Several examples are provided to illustrate our theoretical results.

Symmetry and symmetric transformations in mathematical imaging

The article delves into the intricate relationship between symmetry and mathematical imaging, spanning various mathematical disciplines. Symmetry, a concept deeply ingrained in mathematics, manifests in art, nature, and physics, providing a powerful tool for understanding complex structures. The paper explores three types of symmetriesreflection, rotational, and translationalexemplified through concrete mathematical expressions. Evariste Galoiss Group Theory emerges as a pivotal tool, providing a formal framework to understand and classify symmetric operations, particularly in the roots of polynomial equations. Galois theory, a cornerstone of modern algebra, connects symmetries, permutations, and solvability of equations. Group theory finds practical applications in cryptography, physics, and coding theory. Sophus Lie extends group theory to continuous spaces with Lie Group Theory, offering a powerful framework for studying continuous symmetries. Lie groups find applications in robotics and control theory, streamlining the representation of transformations. Benoit Mandelbrots fractal geometry, introduced in the late 20th century, provides a mathematical framework for understanding complex, self-similar shapes. The applications of fractal geometry range from computer graphics to financial modeling. Symmetrys practical applications extend to data visualization and cryptography. The article concludes by emphasizing symmetrys foundational role in physics, chemistry, computer graphics, and beyond. A deeper understanding of symmetry not only enriches perspectives across scientific disciplines but also fosters interdisciplinary collaborations, unveiling hidden order and structure in the natural and designed world. The exploration of symmetry promises ongoing discoveries at the intersection of mathematics and diverse fields of study.

On mathematical problems of two-coefficient inverse problems of ultrasonic tomography

This paper proves the theorem of uniqueness for the solution of a coefficient inverse problem for the wave equation in (with two unknown coefficients: speed of sound and absorption. The original nonlinear coefficient inverse problem is reduced to an equivalent system of two uniquely solvable linear integral equations of the first kind with respect to the sound speed and absorption coefficients. Estimates are made, substantiating the multistage method for two unknown coefficients. These estimates show that given sufficiently low frequencies and small inhomogeneities, the residual functional for the nonlinear inverse problem approaches a convex one. This solution method for nonlinear coefficient inverse problems is not linked to the limit approach as frequency tends to zero, but assumes solving the inverse problem using sufficiently low, but not zero, frequencies at the first stage. For small inhomogeneities that are typical, for instance, for medical tasks, carrying out real experiments at such frequencies does not present major difficulties. The capabilities of the method are demonstrated on a model inverse problem with unknown sound speed and absorption coefficients. The method effectively solves the nonlinear problem with parameter values typical for tomographic diagnostics of soft tissues in medicine. A resolution of approximately 2 mm was achieved using an average sounding pulse wavelength of 5 mm.

Challenges in Numerical Solutions of Higher-Dimensional Differential Equations

Differential equations constitute a fundamental tool in modeling various natural phenomena across scientific disciplines such as physics, engineering, and finance. We provide an overview of fractional differential equations, focusing on the computational requirements associated with their numerical solutions from a computer science perspective. We analyze the computational intricacies concerning First-Order Linear ODE, First-Order Nonlinear ODE, Second-Order Linear ODE, Second-Order Nonlinear ODE, Heat Equation (PDE), and Wave Equation (PDE). This comparative assessment delves into the computational demands of solving these equations using differential equation methodologies. While analytical solutions provide deep insights, obtaining numerical solutions, particularly in higher dimensions, remains a persistent challenge. Finite difference methods commonly employed for numerical solutions, In higher-dimensional problems, traditional numerical methods face challenges stemming from an exponential surge in grid points and the consequent demand for substantially decreased time step sizes. This paper explores the challenges posed by higher-dimensional differential equations in numerical solutions. It highlights the infeasibility of finite difference methods in such scenarios and emphasizes the need for innovative numerical techniques capable of efficiently handling the complexities of higher-dimensional differential equations. Overcoming these challenges is crucial for advancing our understanding and modeling capabilities in complex real-world systems governed by differential equations. Continued research efforts strive to develop novel numerical methodologies capable of addressing these challenges, aiming to broaden the scope of solvable higher-dimensional differential equations and expand their application across diverse scientific domains.

Investigation of conditions for preserving global solvability of operator equations by means of comparison systems in the form of functional-integral equations in $\mathbf{C}[0;T

Let $U$ be the set of admissible controls, $T>0$ and it be given a scale of Banach spaces $W[0;\tau]$, $\tau\in(0;T]$, such that the set of constrictions of functions from $W=W[0;T]$ to a closed segment $[0;\tau]$ coincides with $W[0;\tau]$; $F[\cdot;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. For the operator equation $x=F[x;u]$, $x\in W$, we introduce a comparison system in the form of functional-integral equation in the space $\mathbf{C}[0;T]$. We establish that, under some natural hypotheses on the operator $F$, the preservation of the global solvability of the comparison system pointed above is sufficient to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation. This fact itself is analogous to some results which were obtained by the author earlier. The central result of the paper consists in a set of signs for stable global solvability of functional-integral equations mentioned above which do not use hypotheses similar to local Lipschitz continuity of the right-hand side. As a pithy example of special interest, we consider a nonlinear nonstationary Navier–Stokes system in the space $\mathbb{R}^3$.

Evolution of an interacting fermion–antifermion pair in the near-horizon of the BTZ black hole

In this study, we investigate the evolution of a composite system comprising a fermion–antifermion pair engaged in Cornell-type non-minimal interaction in the near-horizon region of a BTZ black hole. Our exploration involves the derivation of an exact solution for the covariant two-body Dirac equation, derived from quantum electrodynamics through the action principle. To commence our analysis, we formulate the relevant equation, resulting in a 4×4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 4$$\\end{document} dimensional matrix equation that governs the relative motion of the fermion–antifermion pair. Notably, we demonstrate that this matrix equation results in an exactly solvable wave equation, enabling us to determine the relativistic frequency modes for this spinless static composite system. Our findings unveil a temporal decay in these modes, with the decay time explicitly dependent on both the inter-particle interaction and the spacetime parameters. Our comprehensive examination extends to a detailed analysis of the system’s evolution, shedding light on the influence of inter-particle interaction on the evolution of a fermion–antifermion pair in the near-horizon region of the BTZ black hole.

DISCRETE GROUP METHOD FOR A MATHEMATICAL MODEL OF THE DIFFUSION IN SWELLING GELATIN

In this article, we will introduce the mathematical model of the diffusion process in a swelling medium that has already been modeled. The main purpose of this paper is introducing a new class of group transformations for solving the nonlinear emulsion equation. Thus, we introduce this method as the discrete group methods based on solvable class orbits. One of the crucial advantages of this method is that a transformation is sought, which reduces the equation being investigated to some standard form for which the methods of integration are known. In developing this approach, one may construct all canonical forms of solvable equations.

Efficient least-squares reverse time migration in TTI media using a finite-difference solvable pure qP-wave equation

Abstract The anisotropic characteristics of underground media play a crucial role in seismic wave propagation and should be accounted for during seismic imaging. The least-squares reverse time migration (LSRTM) method achieves ideal imaging results by suppressing migration artifacts and balancing amplitude. Therefore, pure quasi-P (P-qP) wave equations in tilted transversely isotropic (TTI) media have been widely used to implement LSRTM, owing to their ability to produce stable and noise-free wavefields. However, solving the anisotropic P-qP-wave equations typically necessitates the use of spectral-based methods, making them computationally inefficient, especially in 3D applications. In this study, we first develop a P-qP-wave equation in TTI media that can be efficiently computed through the finite-difference (FD) approach. Numerical tests show that, in comparison to the previous TTI P-qP-wave equation, the newly derived FD solvable TTI P-qP-wave equation yields reasonably accurate and highly efficient wavefield simulations. Then, building on our newly derived wave equation, we derive its adjoint migration and demigration operator to implement TTI LSRTM. Two synthetic examples suggest that the newly presented P-qP-wave TTI LSRTM approach effectively correct for the anisotropy effects, providing high-quality imaging results. Additionally, our approach has superior computational efficiency over the conventional P-qP-wave TTI LSRTM technique based on a hybrid FD pseudo-spectral (HFDPS) solver.