Solutions Of Linear Equations Research Articles

Conformable Triple Sumudu Transform with Applications

One of the important topics in applied mathematics is the topic of integral transformations, due to their importance in electrical engineering applications, including communications in particular, and other sciences. In this work, one of the most important transformations in its three dimensions was presented, which is the triple Sumudu transform, including solving some real-life applications of physics, some of which have not been solved using such an integral transform before. In this work, we extend the Sumudu transform formula to the conformable fractional order, as well as other interesting and significant rules. The general analytical solution of a singular and nonlinear conformable fractional differential equation based on the conformable fractional Sumudu transform is also presented in this paper. The general solutions of several linear and nonhomogeneous conformable fractional differential equations can be obtained using the method we’ve proposed. As a result, our results reveal that our proposed method is an efficient one that can be used for solving all conformable fractional differential equations. The relationship between the Sumudu integral transform and other important and recently proposed integral transforms are also discussed. Finally, the triple Sumudu transform is used to solve boundary value problems, such as the heat equation with boundary values. The triple Sumudu integral transform is also used to solve linear partial integro-differential equations. The transform capability to handle such equations has been proven via its utilization in three applications.

Frobenius method for Mahler equations

Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by using a variant of Frobenius method.

A Kinetic Finite Volume Discretization of the Multidimensional PIDE Model for Gene Regulatory Networks

In this paper, a finite volume discretization scheme for partial integro-differential equations (PIDEs) describing the temporal evolution of protein distribution in gene regulatory networks is proposed. It is shown that the obtained set of ODEs can be formally represented as a compartmental kinetic system with a strongly connected reaction graph. This allows the application of the theory of nonnegative and compartmental systems for the qualitative analysis of the approximating dynamics. In this framework, it is straightforward to show the existence, uniqueness and stability of equilibria. Moreover, the computation of the stationary probability distribution can be traced back to the solution of linear equations. The discretization scheme is presented for one and multiple dimensional models separately. Illustrative computational examples show the precision of the approach, and good agreement with previous results in the literature.

On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations

This work is devoted to the development of methods for constructing asymptotic formulas as x→∞ of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh–Levitan regularity conditions). As a model equation, the fifth-order equation i2p(x)y‴″+p(x)y″‴+q(x)y=λy, along with various behaviors of coefficients p(x),q(x), is investigated. New asymptotic formulas are obtained for the case when the function h(x)=−1+p−1/2(x)∉L1[1,∞) significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied.

Extension of Haar wavelet technique for numerical solution of three-dimensional linear and nonlinear telegraph equations

In this manuscript, a hybrid numerical technique is presented for solving three-dimensional hyperbolic telegraph equations. The proposed technique is based on the Haar wavelet collocation method and the finite difference method. In this technique, the space derivatives are estimated by truncated Haar wavelet series, while the time derivatives are approximated using finite difference method. The method is applied to some linear and non-linear three-dimensional hyperbolic telegraph equations. In the linear case, the resulting algebraic system is solved by the Gauss elimination technique. Whereas in the nonlinear case, the quasi-Newton linearization technique is utilized to remove the nonlinearity of the problem. The maximum absolute errors, root mean square errors and computational rate of convergence are calculated for a different number of collocation points at different time steps of the final time t=1. The stability analysis for the proposed method is also discussed. Overall, the recently devised numerical method boasts straightforward implementation, rapid convergence, and enhanced accuracy in addressing both linear and nonlinear problems.

An asymptotic solution for the SIS epidemic model, taking into account migration and diffusion

The purpose of this work is to propose and investigate a simple and effective model of an epidemic in an animal population that takes into account migration along the plane of both diseased and healthy individuals. Within the framework of this model, the spatial migration of a population is described by introducing both diffusion and advective terms into its equations. Methods. In this paper, a method of many scales was used to find an asymptotic solution to the system of equations of the epidemic. Solutions of auxiliary linear equations of the parabolic type arising during this procedure were found using the Poisson integral. The simplification of the initial system of equations of the model is based on the assumption that the sum of densities of healthy and sick individuals on a single-connected region of large diameter on the plane is constant at the initial moment of time. Results. It is shown that in this case, designed for a slowly changing initial density of sick individuals concentrated inside this area at a considerable distance from its boundaries, the asymptotic solution of the model describes the effect of merging several spatially spaced small outbreaks of the disease into one large outbreak during migration of the entire population as a whole. In particular, for such an initial density obtained by the functional transformation of a Gaussian, a circular plateau is formed over long periods with an effective radius that grows linearly over time. Conclusion. The constructed asymptotic solution of the epidemic model proposed in this paper is simple in form and describes the transfer of the disease on a locally flat area of the earth’s surface without the use of numerical methods. This solution is convenient when describing the migration of a sick population under the influence of flooding, forest fire, man-made disaster with contamination of the area, etc.

Continuous Solutions of Linear Functional Equations on Piecewise Smooth Curves in Mathematical Models of Boundary Value Problems with a Shift

Continuous Solutions of Linear Functional Equations on Piecewise Smooth Curves in Mathematical Models of Boundary Value Problems with a Shift

On asymptotic behavior of solutions of multivariate linear stochastic differential equations with multiplicative noise

On asymptotic behavior of solutions of multivariate linear stochastic differential equations with multiplicative noise

On the upper bounds for the distance between zeros of solutions of a first-order linear neutral differential equation with several delays

<p>This work is devoted to studying the distribution of zeros of a first-order neutral differential equation with several delays</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left[y(t)+a(t)y\left(t-\sigma\right)\right]'+ \sum\limits_{j = 1}^n b_j(t)y\left(t-\mu_j\right) = 0, \quad \quad \quad t \geq t_0. \end{equation*} $\end{document} </tex-math></disp-formula></p><p>New estimations for the upper bounds of the distance between successive zeros are obtained. The properties of a positive solution of a first-order differential inequality with several delays in a closed interval are studied, and many results are established. We apply these results to a first-order neutral differential equation with several delays and also to a first-order differential equation with several delays. Our results for the differential equation with several delays not only provide new estimations but also improve many previous ones. Also, the results are formulated in a general way such that they can be applied to any functional differential equation for which studying the distance between zeros is equivalent to studying this property for a first-order differential inequality with several delays. Further, new estimations of the upper bounds for certain equations are given. Finally, a comparison with all previous results is shown at the end of this paper.</p>

ON THE BEHAVIORS OF SOLUTIONS IN LINEAR NONHOMOGENEOUS DELAY DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

This paper deals with the behaviors of solutions for linear nonhomogeneous delay differential equations. In this study, a periodic solution, an asymptotic result and a useful exponential estimate of the solutions are established. Our results are obtained by the use of real roots of the corresponding characteristic equation.

About an Unsolved Problem of Stabilization by Noise for Difference Equations

The paper is devoted to the effect of “stabilization by noise”. The essence of this effect is that an unstable deterministic system is stabilized by stochastic perturbations of sufficiently high intensity. The problem is that the effect of “stabilization by noise”, well-known already for more than 50 years for stochastic differential equations, still has no analogue for stochastic difference equations. Here, a corresponding hypothesis is formulated and discussed, the truth of which is illustrated and confirmed by numerical simulation of solutions of stochastic linear and nonlinear difference equations. However, a problem of a formal proof of this hypothesis remains open.

An endeavor from the Glowinski-Le Tallec splitting for approximating the solution of Kawarada equation

This paper studies an extended application of the Glowinski-Le Tallec splitting for approximating solutions of linear and nonlinear partial differential equations. It is shown that the three-level, six-component operator decomposition, originally designed for Lagrangian optimizations, provides a stable second-order operator splitting approximation for the solutions of evolutional partial differential equations. It is also found that the Glowinski-Le Tallec formula not only provides an effective enhancement to conventional two-level, four-component ADI and LOD methods, but also introduces a flexible way for constructing multi-parameter operator splitting strategies in respective spaces where broad spectrums of mathematical models may exist for important natural phenomena and applications. The extended operator splitting is utilized for solving a singular and nonlinear Kawarada problem satisfactorily. Multiple simulation results are presented.

THE CHAIN METHOD FOR SOLVING A FINITE-ORDER LINEAR DIFFERENCE EQUATION AND SOME OF ITS APPLICATIONS

A scheme of the chain method for solving a finite linear difference equation is given in this paper, and a formula for this equation’s general solution of is given. As a result, the formula for the general solution of a difference equation with constant coefficients is given. This formula depends entirely only on the coefficients of this equation. Considered solutions of linear differential equations in the form of a generalized power series, the coefficients of which are found by the chain method. As a result of permuting the elements of the power series, the solution of the equation contains a new function, namely: a hypergeometric function of fractional order.

ASYMPTOTIC REPRESENTATION OF SOLUTIONS CLOSE TO LINEAR DIFFERENTIAL EQUATIONS OF THE THIRD ORDER

In the work, using the a priori properties of the Pω(λ0) class of solutions, the conditions for the existence of one class of solutions in the binomial non-autonomous of a differential equation of the third order with a nonlinearity close in some sense to linear, in the critical case, namely when λ0 = ±∞. Asymptotic at t↑ω (ω ≤ +∞) images for such solutions and their derivatives of the first and second order in the case λ0 = ±∞. Proven for of the nonlinear equation, the lemma and the theorem are transferred to linear differential equations of the third order with asymptotically small coefficients. The transferred results do not contradict, and to some extent, complement known results regarding the asymptotic behavior of solutions of linear differential equations of the third order.

Estimates of matrix solutions of operator equations with random parameters under uncertainties

We investigate problems of estimating solutions of linear operator equations with random parameters under conditions of uncertainty. We establish that the guaranteed rms estimates of the matrices are found as solutions of special optimization problems under certain observations of the system state. As the output signals of the system, we have observations that are described by linear functions from the solutions of such equations with random right-hand sides, which have unknown second moments. Under the condition that the observation second moments of the right-hand parts and errors belong to certain sets, it is proved that the guaranteed estimates are expressed through solutions of operator equation systems. When the linear operator is given by the scalar product of rectangular matrices, a quasi-minimax estimate and its error are constructed. It is shown that the quasi-minimax estimation error tends to zero when the number of observations tends to infinity. An example of calculating the guaranteed rms estimate of the matrix's trace, which is a solution of a matrix equation with a random parameter, is given.

Growth and value distribution of linear difference polynomials generated by meromorphic solutions of higher-order linear difference equations

In this article, we investigate the relationship between growth and value distribution of meromorphic solutions for the higher-order complex linear difference equations $$ A_n(z)f(z+n)+\dots+A_1(z)f(z+1)+A_0(z)f(z)=0 \quad \text{and } =F(z), $$ and for the linear difference polynomial $$ g(z)=\alpha_n(z)f(z+n)+\dots+\alpha_1(z)f(z+1)+\alpha_0(z)f(z) $$ generated by \(f(z)\) where \(A_j(z)\), \(\alpha_j(z)\) (\(j=0,1,\ldots,n\)), \(F(z)\) \((\not\equiv0)\) are meromorphic functions. We improve some previous results due to Belaidi, Chen and Zheng and others.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/84/abstr.html

Nonlinear forced vibration of a nanobeam resting on Winkler-Pasternak elastic foundation and subjected to a mechanical impact

The nonlinear governing equations of nanobeam taking into consideration its curvature, resting on an elastic Winkler-Pasternak foundation and based on non-local Euler-Bernoulli beam theory is analyzed. The equation of motion and the boundary conditions are modeled within the framework of a simple supported nanobeam which accounts the presence of a mechanical impact and nonlinear von-Karman strain. The resulting nonlinear differential equations are reduced to only one differential equation which is studied by means of the Optimal Auxiliary Functions Method (OAFM). An explicit analytical solution is proposed for a complex problem. The main quality of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution for the initial linear equation and the form of nonlinear term calculated from the above solution of the linear equation. The convergence-control parameters present in the auxiliary functions are evaluated by a rigorous mathematical procedure. The obtained solutions are in very good agreement with the numerical solution.

Differential Transform Method for Solving Volterra q-Integral Equations

In this paper, Volterra q-integral equations are solved by using the method of q-differential transformation. Exact solutions of linear and nonlinear q-integral equations have been investigated. To illustrate the method, several problems are discussed for the effectiveness and performance of the proposed method.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

Numerical solutions of linear time-fractional advection-diffusion equations with modified Mittag-Leffler operator in a bounded domain

Fractional advection-diffusion equations have demonstrated to be a powerful tool in modeling complex anomalous diffusion in applied science. In this paper, we studied novel linear time-fractional advection-diffusion equations associated with an extension of Mittag-Leffler fractional derivative operator. A useful feature of the used extension is to address the limitations of the Mittag-Leffler fractional derivative model. We, mainly, proposed a numerical approach to provide approximate solutions to linear time-fractional advection-diffusion equations with the studied extended fractional derivative operator. The suggested approach is based on discretizing the studied models with respect to spatio-temporal domain using uniform meshes. A new type of solutions for the studied models was generated numerically using the proposed approach. Besides, a comparative study was conducted to verify the accuracy and feasibility of the proposed approach.