Solvability Of Equations Research Articles

Gross misinterpretation of a conditionally solvable eigenvalue equation

We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions for particular values of a model parameter. From these solutions some researchers have derived a variety of predictions like allowed angular frequencies, allowed field intensities and the like. We also solve the eigenvalue equation numerically by means of the variational Ritz method and compare the resulting eigenvalues with those provided by the truncation condition. In this way we prove that those physical predictions are merely artifacts of the truncation condition.

New classes of hyperbolic‐cotangent–type systems of difference equations solvable in closed form

In the case of homogeneous linear difference equations with constant coefficients, only a relatively small part of the equations is practically solvable. This is connected to impossibility to solve all polynomial equations of degree bigger than four. In the case of nonlinear difference equations and systems the situation is even worse. So it is of interest to find new classes of practically solvable difference equations and systems. Recently, there have been presented several classes of practically solvable hyperbolic‐cotangent–type systems of difference equations. Here, we present a new class of solvable systems of this type. It is a bit surprising that all the systems in the class are practically solvable, especially since several special cases are connected to some polynomials of degree eight.

Non-Dimensional Groups, Film Thickness Equations and Correction Factors for Elastohydrodynamic Lubrication: A Review

Apart from complex numerical models to predict the tribological behavior of elastohydrodynamically lubricated contacts, non-dimensional similarity groups and analytically solvable proximity equations can be used to estimate integral fluid film parameters. Based upon the pioneering work presented by Dowson and Higginson as well as Blok and Moes, these approaches have been continuously improved over the years by modifications or correction factors to capture different contact geometries (line-, point- or elliptical contacts) as well as to include fluid compression, thermal, non-Newtonian, starvation or roughness effects. Consequently, this review article aims at systematically reviewing these modifications/corrections and discussing their applicability as well as limitations before presenting some recommendations for future research activities.

Weak Solvability of Equations Modeling Steady-State Flows of Second-Grade Fluids

We prove the existence of continuous weak solutions of the nonlinear equations describing steady-state flows of second-grade fluids in a bounded three-dimensional domain under the no-slip boundary condition. A weak solution is found using the Galerkin method with special basis functions constructed with the help of a perturbed Stokes operator. An energy inequality for the resulting solution is derived.

Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations

Here, we shed light on the fractional linear and nonlinear Klein–Gorden partial differential equations via Fractional Shifted Legendre Tau Method. With this objective, the operational matrices of fractional-order shifted Legendre functions (FSLFs) are derived and combined with the Tau method to convert the fractional-order differential equations to a system of solvable algebraic equations. The validity and the efficiency of the operational matrices are tested. Our findings yield an affirmative consequence, indicating applicability of the proposed method for nonlinear equations appearing in science and engineering.

On the solvability of hypersingular equation of peridynamics

On the solvability of hypersingular equation of peridynamics

Effect of temperature fluctuation on the localized pattern of action potential in cardiac tissue

Based on the improved FitzHugh–Nagumo myocardial model driven by a constant external current, the effect of temperature fluctuation in a network of electrically coupled myocardial cells are investigated through analytical and numerical computations. Through the technique of multiple scale expansion, we successfully reduced the complex nonlinear system of equations to a more tractable and solvable nonlinear amplitude equation on which the analysis of linear stability is performed. Interestingly from this analysis, a plot of critical amplitude of action potential versus wave number revealed the growth rate of modulational instability (MI) is an increasing function of the thermoelectric couplings; T^{(1)} and T^{(2)}, under fixed conditions of nonlinear electrical couplings. In order to verify our analytical predictions through the study the long-time evolution of the modulated cardiac impulses, numerical computation is finally carried out. Numerical experiment revealed the existence of localized coherent structures with some recognized features of synchronization. Through the mechanism of MI, changes in thermoelectrical couplings promote wave localization and mode transition in electrical activities in the cell lattice. Results could provide new insights in understanding the underlying mechanism of the manifestation of sudden heart disorder subjected to heavily temperature fluctuation.

Solvability of Pseudoparabolic Equations with Non-Linear Boundary Condition

The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann–Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin’s method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev ’s attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates.A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann–Dirichle boundary condition.

On Solvability of Integro-Differential Equations

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.

Solvability for indefinite mean‐field stochastic linear quadratic optimal control with random jumps and its applications

SummaryIn this article, we study the open‐loop and closed‐loop solvability for indefinite mean‐field stochastic linear quadratic (MF‐SLQ) optimal control problem and its application in finance, where the controlled stochastic system is driven by a Brownian motion and a Poisson random martingale measure and also disturbed by some stochastic processes. The intrinsic property of stochastic systems results in the inequivalence of those two solvabilities, which is different from deterministic case. Based on a well‐posedness result of the problem, it is shown that the uniform convexity of cost functional is sufficient for the open‐loop solvability of the problem. By a matrix minimum principle, the necessity condition of regular solvability for a decoupled Riccati equation is established, meanwhile, the closed‐loop solvability is turned out to be equivalent to the regular solvability of Riccati equations with some constraints on the solutions of disturbances equations, moreover, the optimal closed‐loop strategy is characterized by regular solutions of Riccati equations and adapted solutions of disturbances equations. And then a mean‐variance model is considered to solve optimal portfolio selection strategy problem of the insurance company with liability. Our study fills a gap in the field of solvability for mean‐field stochastic optimal control with random jumps and provides a different way for solving mean‐variance portfolio selection model.

Solvability and optimal controls of non-instantaneous impulsive stochastic fractional differential equation of order q ∈ (1,2)

In this article, we study the stochastic fractional optimal control problem for a system governed by a class of non-instantaneous impulsive stochastic fractional differential equation in the infinite-dimensional spaces. We utilize the fractional calculus, stochastic analysis theory and a suitable fixed point approach to present the solvability of the stochastic fractional system. Then, the existence of optimal state-control pairs of the Lagrange problem is derived without uniqueness of solutions of the stochastic system. Finally, the main results are validated with the aid of an example.

Solvability for perturbations of a class of real vector fields on the two-torus

Let L=∂t+a(x)∂x be a real vector field defined on the two-dimensional torus T2, where a is a real-valued and smooth function on T1. We deal with the global solvability of equations in the form Lu+pu=f, where p,f∈C∞(T2). Solvability to the equation Lu=f is well-understood. We show that a perturbation of zero order may affect the global solvability of L; we may maintain, gain or lose solvability by adding a perturbation. This phenomenon is linked to the order of vanishing of the coefficient a of L. We obtained results in the class of smooth functions on T2 and, also, in the space of Schwartz distributions D′(T2).

Solving Nonlinear p-Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point Theorem

Recently theory of p-adic wavelets started to be actively used to study of the Cauchy problem for nonlinear pseudo-differential equations for functions depending on the real time and p-adic spatial variable. These mathematical studies were motivated by applications to problems of geophysics (fluids flows through capillary networks in porous disordered media) and the turbulence theory. In this article, using this wavelet technique in combination with the Schauder fixed point theorem, we study the solvability of nonlinear equations with mixed derivatives, p-adic (fractional) spatial and real time derivatives. Furthermore, in the linear case we find the exact solution for the Cauchy problem. Some examples are provided to illustrate the main results.

A Direct Solving Approach for High-Order Power Amplifier Matching Network Design

In this article, a direct solving method for power amplifier (PA) matching network design is explored from a theoretical perspective. Commonly, the input impedance function of a multivariable matching network is a transcendental equation and it is unsolvable. In the proposed new method, this problem can be solved to some extent by transforming the transcendental equations into solvable linear equations, where the unknown coefficients can be calculated easily. A family of impedance functions is obtained if the order of matching network is higher than the number of functions, and the best one can be selected according to the practical demand in PA design. As a design example, a triband PA with arbitrary frequency points (0.9/2.3/3.5 GHz) is implemented to validate the proposed approach. The calculation results show that the proposed method can solve the matching problem of high-order circuits. And the simulation and experiment results agree with the calculation results very well, which verified the validity of the proposed method.

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.

Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion

We are involved in this study with hybrid functions consisting of Taylor polynomials and block-pulse functions and use them as basis functions to achieve the numerical solution of stochastic evolution equation with fractional Brownian motion (FBM). First, we compute stochastic operational matrix based on hybrid Taylor block-pulse functions (HTBPFs) that this operator able us obtain an applicable procedure for solving stochastic integral equations (SIEs) and stochastic differential equations (SDEs) such as stochastic evolution equation. Then, we use this operator and HTBPFs ordinary operational matrix to convert solving stochastic evolution equations with FBM into solving a system of easily solvable algebraic equations. Under some mild conditions we prove that our proposed method is convergent when M, N → ∞, where N and M are the order of block-pulse functions and Taylor polynomials, respectively. Finally, we utilize the mentioned method for solving two test problems that their exact solution is available. Comparison obtained approximate solution with the exact solution demonstrate that the values of absolute error can be ignored and thus obtained solution has a good degree of accuracy. Furthermore, the theoretical discussions and numerical examples confirm that by increasing the values of N and M, the approximate solution tends to the exact solution.

Circular Slit Maps of Multiply Connected Regions with Application to Brain Image Processing

In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is O((M + 1)n), where M+1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require O((M+1)^3 n^3) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented.

Solvability of Impulsive Nonlinear Partial Differential Equations with Nonlocal Conditions

The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem.

The symmetry analysis of electrostatic micro-electromechanical system (MEMS)

The model of electrostatic Micro-Electromechanical System (MEMS) is considered without/with an external pressure. The model represents nonlinear elliptic or parabolic problem due to the steady or non-steady state problem, respectively. To obtain exact solutions in an explicit form, the symmetry analysis is considered. With symmetries, the transformations are obtained and by means of these transformations, solvable equations are hold. The obtained results have a major role in the literature so that the considered equation is seen in a large-scale applications.