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In this paper, a combination of boundary value problems for the nonhomogeneous analytic equations have been studied. The aim of this paper is to find a solution for the Schwarz-(Dirichlet-Neumann) problem and obtain its solvability conditions. [ABSTRACT FROM AUTHOR]

Path planning within three-dimensional environments is of great importance in contemporary time due to the introduction of robots in many works. Finding the shortest path reduces fuel and time. In this paper a new algorithm is proposed to plan the paths within 3D environments. The algorithm was tested in two ways: simulation and practical. A multi-propeller vehicle is built that can track the resulting path from the proposed algorithm. The spherical shape of the robot, and obstacles is the basic of the work of the proposed algorithm. This does not mean that the algorithm cannot handle other forms of obstacles. Any other shape can be represented by a set of intersecting spheres. The resulting path of this proposed algorithm consists of two pieces that can be repeated a number of times until reaching the target point. The first piece is a straight path connecting the current position of the robot with a point located on the edge of the obstacle. The second piece is a path around the obstacle connecting two points away from the obstacle with a distance equal to the radius of the robot. All points are calculated by equations derived in this paper. The proposed algorithm was compared with two other algorithms. The results proved the superiority of the proposed algorithm in terms of path length and smoothness of the path. [ABSTRACT FROM AUTHOR]

The aim of this paper is to create/proposea new hybrid root finding algorithm to solve the given transcendental equations. The algorithm proposed in this paper is built on the trigonometrical algorithm using hyperbolic tangentfunction to find a root. Couples of numerical examples and one sample computations are presented to explain the proposed algorithms, efficiency and accuracy. Implementation of the proposed algorithms is presented in a mathematical software tool Maple. [ABSTRACT FROM AUTHOR]

For a number of years, numerical integration of the aircraft's equation of motion seemed very well established. Virtually no new studies have been dedicated to this topic, while a significant amount of efforts have been dedicated to improving optimization algorithms, and to implementing optimization algorithms in various design and analysis procedures. However, the quality of the results in many optimization procedures depends upon the availability of high amounts numerical data which is often obtained from numerical simulations, hence availability of faster integration algorithms became once again of interest. The current paper discusses a method of utilizing a short arcs analytical solution of airplane's longitudinal channel equations to accelerate the numerical integration. [ABSTRACT FROM AUTHOR]

In this paper we present new results about the existence and regularity properties of a class of parabolic equations where lower order terms appear. We prove that the interplay between coefficients produces a strong regularizing effect since there exist solutions that become immediately bounded despite the lack of regularity of the initial data and the forcing terms which are only summable functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the Camassa-Holm equation with a loaded term in the class of periodic functions. The main result of this work is a theorem on the evolution of the spectral data of the weighted Sturm-Liouville operator whose potential is a solution to the periodic Camassa-Holm equation with a loaded term. The obtained equality (9) allows us to apply the method of the inverse spectral transform to solve the Cauchy problem for the periodic Camassa-Holm equation with a loaded term. [ABSTRACT FROM AUTHOR]

The two-dimensional shallow water equations with a particular bottom and the Coriolis’s force f = f0 + Ωy are studied in this paper. The main goal of the paper is to describe all invariant solutions for which the reduced system is a system of ordinary differential equations. For solving the systems of ordinary differential equations we use the sixth-order Runge-Kutta method. [ABSTRACT FROM AUTHOR]

The authors of this manuscript have worked to investigate the existence of the time periodic weak solution Quasi-linear Chafee-Infante Equation with periodic initial conditions and Neumann boundary conditions. Since the equation our paper is degenerate, so we first need to establish the regular problem. By using Moser iteration technique, we establish a priori upper bound of the weak solution by using a good method that is called the Moser iteration technique. Then by usig the way of contradiction, we get a priori lower bound of the weak solution. This paper is based mainly on the fixed point theorem of infinite spaces, where we used the Leray–Schauder theory to investigate the existence of a non-trivial non-negative time periodic weak solution. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the Beltrami operator for the unit disc 픻. In this work, we investigate the solvability of the Schwarz problem for higher order Beltrami equation. In the unit disc, the explicit solution is given for this equation. [ABSTRACT FROM AUTHOR]

In this paper, we find a solution to the "digital twin" problem i.e., how to completely or exactly represent an object so that we can represent it digitally, where the model behaves and looks like the real-world object. This solution not only addresses the digital twin problem but also enables a generic representation of the Universe. Once we solve this problem we dwell into the phenomenon of "Being" and quantify the quality of "Being" or "Existence". A theory is provided as to how the Universe originated from Nothingness. Furthermore, we represent the Universe through a matrix-based equation. The interpretation of this equation explains how "Existence", creates, sustains, destroys, and controls the Universe. Additionally, the law of action and reaction is explained, or the natural consequence of any action performed by a subject. [ABSTRACT FROM AUTHOR]

In this paper, we first explore what is the phenomenon of "Being", and then we calculate the value of the quality of "Being" or "Existence". A derivation of the powers and states of "Existence" follows. Then we explain what the material cause of all things in the universe is or what it is made of. The material cause is surprisingly mathematical in nature. Then a theory is provided as to how the Universe originated from Nothing. Finally, an equation is derived which describes the state of the Universe at any given instant of time, which traces its origins from "Existence". [ABSTRACT FROM AUTHOR]

This paper deals with an analytical technique used to estimate and resolve the steady-state Frank-Kamenetskii problem. A homotopy perturbation method is applied to solve the nonlinear steady-state thermal explosion in a vessel. Additionally, using the Matlab software, the numerical simulation of the problem is described in this study. The interior temperature of the vessel is estimated and analysed to variations in the Frank-Kamenetskii and geometry parameters. A graphic representation of a vessel's temperature result is provided to demonstrate the solution, and theoretical discussion is had. The results of the analytical and numerical methods show good agreement. [ABSTRACT FROM AUTHOR]

In this paper we define a risk model with stochastic premium process, such that the counting processes are not independent. We consider a special case of bivariate counting process. Then we derive equations for the non-ruin probability in infinite time and analyze the case of exponentially distributed claims. [ABSTRACT FROM AUTHOR]

The paper presents a new mathematical model of the convection-diffusion process with 'memory along the flow path'. It is described by one-dimensional initial-boundary value problem with a fractional derivative along the characteristic curve of convection operator. The constructed model satisfies the local and global conservation laws. The finite-difference approximation of the problem is constructed on the base of Lagrange approach. The stability and discrete conservation laws are proven for algorithmic realization of this approximation. [ABSTRACT FROM AUTHOR]

In this paper some ill-posed boundary-value problems (BVPs) for three - dimensional partial differential equations are studied. The situation with them is rather surprising and there is no general understanding even more than sixty years after their statement given by Murray Protter. These problems are multidimensional analogues of classical BVPs on the plane and intuitively the initial expectations was that their properties would be similar. Unexpectedly, it turned out that unlike the two-dimensional variants, the Protter-Morawetz problems are not well posed. The generalized solution is uniquely determined, but it may have a strong singularity at an isolated boundary point even for infinitely smooth right-hand side. Also, the adjoined problem has an infinite number of smooth solution in the kernel. In the present paper such ill-posed problems for 3-D Gellerstedt equation with lower order terms under multidimensional Protter condition are studied. In addition to new results, we also make a survey of the known results concerning the Protter-Morawetz problems for both Tricomi-type equations and Keldysh-type equations. [ABSTRACT FROM AUTHOR]

In research paper, the queuing model is constructed having "N-Serial service channels in which the customers arrive in Poisson stream", are served exponentially and selected by SIRO discipline. The concepts of feedback, balking and reneging are incorporated at each stage of the queuing model. It has been considered here that the customers may fix a few service channel from outside directly and, may depart queuing model before or after receiving the service. After describing the queuing model, we write its equations and find their solutions for unlimited as well as limited capacity. The marginal probability and average length of the waiting line of the system have been determined for infinite capacity whenever the SIRO discipline is replaced with FCFS for knowing the achievement of the system. Particular cases have been derived here. [ABSTRACT FROM AUTHOR]

The paper investigates a linear fractional nonhomogeneous discrete equation Δ a y (n + 1) = a y (n) + ω (n) , n = n 0 , n 0 + 1 , ... where Δα is the fractional α-order difference, α > 0, n is an independent variable, y: {n0, n0 + 1, ...} is an unknown function, a ∈ ℝ, and ω: {n0n0 + 1,...} → ℝ. The asymptotic behaviour of particular solutions caused by the function ω is studied. It is shown that, under suitable conditions, there exists a particular solution of the equation in question characterized by an exponential-type function. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the optimization algorithms Adam, RMSprop, L-BFGS and SGD with momentum on the solution of Fractional Laplacian Equation (FLE) by physics-informed neural networks is analysed when considering two different analytical solutions, one smooth and one non-smooth. The influence of the optimization method, the smoothness of the analytical solution and the network configuration on the accuracy of the predicted solution is discussed in detail. [ABSTRACT FROM AUTHOR]

The paper considers a linear fractional discrete equation Δ α x (n + 1) = λ x (n) = δ (n) , n = 0 , 1 , ... where Δα is the fractional α-order difference, α > 0, λ ∈ ℝ and δ: {0, 1, ...} → ℝ. A problem is considered of the existence of a solution x: {0, 1, ...} → ℝ satisfying |x(n)| < M, n = 0, 1, ..., where M is a constant. This problem is also considered for an equation Δ α x (n + 1) = λ (n) x (n) + δ (n , x (n) , x (n − 1) , ... , x (0)) , n = 0 , 1 , ... , , where λ : { 0 , 1 , ... } → ℝ , δ : { 0 , 1 , ... n } × ℝ × ℝ × ... ℝ ︸ n + 1 → ℝ , generalizing the previous one. [ABSTRACT FROM AUTHOR]

This paper deals with radial and non-radial solutions for local and non-local Liouville type equations. At first non- degenerate and degenerate mean field equations are studied and radially symmetric solutions to the Dirichlet problem for them are written into explicit form. The Cauchy boundary value problem for nonlinear Laplace equation with several exponential nonlinear- ities is considered and C2 smooth monotonically decreasing radial solution u(r) is found. Moreover, u(r) has logarithmic growth at ∞. Our results are applied to the differential geometry, more precisely, minimal non-superconformal degenerate two dimensional surfaces are constructed in R4 and their Gaussian, respectively normal curvature are written into explicit form. At the end of the paper several examples of local Liouville type PDE with radial coefficients which do not have radial solutions are proposed. [ABSTRACT FROM AUTHOR]

This paper proposes a stabilization method to approximate analytical solutions of magnetohydrodynamics (MHD) equations. The method adds two modular grad-div steps into fully-discrete finite element MHD solver. The main idea in these intrusive steps is to penalize the divergence of the velocity/magnetic fields both in L2 and H1-norms. The paper confirms the optimal convergence of the method, and gives numerical experiments which reveal positive effect of the method as in the usual grad-div stabilization. [ABSTRACT FROM AUTHOR]

In this paper, vibrations in a visco-poroelastic cylinder over a heterogeneous layer in the presence of an initial stress are investigated. Employing the Biot's theory, the governing equations are formulated. Frequency is computed as a function of wavenumber for fixed initial stress and heterogeneous parameter values. For illustration purpose, two solids are considered and the results are presented graphically. [ABSTRACT FROM AUTHOR]

In the paper is considered a fractional discrete equation ∑ π = 1 s Δ β π z (k + 1) = G k (k , z (k) , ... z (k 0)) , k = k 0 , k 0 + 1 , ... where Δβπ, βπ > 0, π = 1, ...s, are the βπ order fractional differences, Gk:{k} × ℝ k-k0+1 → ℝ, k0 ∈ ℤ, k ∈ ℤ , k ≥ k0 and z: {k0, k0 + 1...} → ℝ. Sufficient conditions are given for the existence of bounded solutions satisfying inequalities b(k) < z(k) < c(k), ∀k ≥ k0 where b and c are real functions satisfying b(k) < c(k). An application is considered to an equation with several fractional differences ∑ π = 1 s Δ β π z (k + 1) = ξ z (k) + σ (k) , k = k 0 , k 0 + 1 , ... where ξ ∈ ℝ and σ:{k0, k0 + 1...} → ℝ. It is proved that there exists a bounded solution satisfying the inequality |z(k)| < L, k = k0, k0 + 1, ..., for a constant L. [ABSTRACT FROM AUTHOR]

In the paper we consider a two-dimensional linear non-homogeneous system of discrete equations y 1 (k + 1) = a y 1 (k) + p y 2 (k) + g 1 (k) , y 2 (k + 1) = − q y 1 (k) + a y 2 (k) + g 2 (k) , where k = k0, k0 + 1,... with k0 a fixed integer, a, p > 0, q > 0 are real constants and gi: {k0, k0 + 1,...} ℝ, i = 1, 2 are given functions. Sufficient conditions are derived guaranteeing the existence of a solution y(k) = (y1(k), y2(k)), k = k0, k0 + 1, ... satisfying α y 1 2 (k) + β y 2 2 (k) < M , where M, α and β are positive fixed constants such that αp = βq. [ABSTRACT FROM AUTHOR]

In this short paper, we study the Cauchy problem associated with the forced time-fractional relaxation-oscillation equation with distributed order. We employ the Laplace transform technique to derive the solution. Additionally, for the scenario without external forcing, we focus on density functions characterized by a single order, demonstrating that under these conditions, the solution can be expressed using two-parameter Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

The paper considers generalized Rayleigh equation with control. A parametric analysis of the self-oscillating properties of this equation with zero control is carried out both for the case of a linear and for the case of a non-linear conservative function. Sufficient conditions for complete controllability of the corresponding system are obtained. In connection with the discontinuous dependence of the controllability set on the parameter, the corresponding value of the parameter is estimated, and the corresponding lower bound is constructed for the bounded controllability set. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior and oscillation criteria of neutral differential equations of Emden-Fowler type of third order are studied. Where some conditions were formulated to ensure oscillation for all solutions of these equations. The obtained conditions can be generalized to higher order delay differential equations of Emden-Fowler type. The obtained results showed that the Emden-Fowler type in the neutral differential equation plays a major role in the presence or absence of the oscillation property, unlike other types of neutral differential equations. Some examples are presented to illustrate and apply the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear Rosenau-Hyman equation with time dependent variable coefficients is considered for inves-tigating its invariant properties, exact solutions and conservation laws. Using Lie classical method, we derive point symmetries admitted by the governing equation. Symmetry reductions are performed for each components of optimal set. Also nonclassical approach is employed to find some additional supplementary symmetries and corresponding symmetry reductions are performed. Later three kinds of exact solutions of considered equation are presented graphically for different parameters. In addition, local conservation laws are constructed for the considered equation by multiplier approach. [ABSTRACT FROM AUTHOR]

In the present paper a Coupled KdV equation is transformed using a new scaling technique involving one of the dependent variables u in potential form and other dependent variable v in square form. A regular perturbation series is computed in a non trivial and non-cumbersome way. The analysis interconnects u and v. As a result, one of the equations can be recast into linearly transformed KdV equation and the other into modified KdV equation. Since both have Lax matrix pairs, they are integrable. Also different situations come up due to equality of convection and nonlinear terms are discussed in the analysis of solutions. [ABSTRACT FROM AUTHOR]

The time-fractional Black-Scholes (TFBS) model is often used in the modeling of the fractional structure observed in financial markets. In this paper we propose a high-order compact numerical method for the solution of the pricing equation of the TFBS model. The proposed method uses a fourth-order compact three-point approximation in space and an implicit Euler discretization in time combined with an extrapolation technique. Numerical examples are carried out to show the merits of the method for pricing European and knock-out options under the TFBS model arising in the financial market following the discovery of fractals. [ABSTRACT FROM AUTHOR]

The definition of balancing and cobalancing numbers are generalized to sequence balancing and cobalancing numbers as well as to almost balancing and cobalancing numbers. The sequence cobalancing numbers corresponding to the sequence {n + 1} for n > 0 asks for the existence of all pronic numbers which are 1 more than a triangular number. Similarly, the almost cobalancing numbers of first kind asks for all pronic numbers which are 1 less than a triangular number, whereas the second kind asks for the pronic numbers which are 1 more than a triangular number. Though the solutions of almost cobalancing numbers have been investigated using the theory of Pell's equation, the sequence cobalancing numbers for the sequence {n + 1} was not explored. In this paper, we study the problem of investigating the existence of all pronic and triangular numbers differing by 1 and we represent the solutions in terms of balancing and cobalancing numbers. However, our way of approaching to the solutions of Diophantine equations does not involve the theory of Pell's equation. [ABSTRACT FROM AUTHOR]

In this investigation, the generalized ( G ′ G 2 ) –expansion method is proposed and applied to the generalized Zakharov-Kuznetsov (ZK) equation with variable coefficient, which exists in many scientific fields like, plasma material science, and optical fiber. Further, our aim in this paper is to achieve the closed form solutions of ZK equation. The newly presented solutions are of hyperbolic, trigonometric, and rational functions. The dynamical representation of the solutions are shown as annihilation of three–dimensional kink waves, and multi-soliton waves. [ABSTRACT FROM AUTHOR]

In this paper, we used the generalized-improved ( G ′ G) expansion method to get the traveling wave solutions of the Schamel equation. The obtained solutions are of the hyperbolic functions, trigonometric functions and rational functions. The dynamical structures of the obtained solutions are provided through the three dimensional plots and shock wave profiles are shown. [ABSTRACT FROM AUTHOR]