Solutions Of Nonlinear Equations Research Articles

(G'/G)-EXPANSION METHOD TO SEEK TRAVELING WAVE SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PDES ARISING IN NATURAL SCIENCES

The (G'/G)-expansion method with the aid of symbolic computational system can be used to obtain the traveling wave solutions (hyperbolic, trigonometric and rational solutions) for nonlinear time-fractional evolution equations arising in mathematical physics and biology. In this work, we will process the analytical solutions of the time-fractional classical Boussinesq equation, the time-fractional Murray equation, and the space-time fractional Phi-four equation. With the fact that the method which we will propose in this paper is also a standard, direct and computerized method, the exact solutions for these equations are obtained.

Examination of dark and bright solitons of (2+1)-dimensional Kundu-Mukherjee-Naskar equation via unified solver technique

In this study, dark and bright solitons of the (2+1)-dimensional Kundu-Mukherjee-Naskar equation are constructed with unified solver in terms of He's variations method. In accordance with basic properties of proposed technique, some dark and bright solitons are obtained. Moreover, giving specific values to the achieved solutions, 2D and 3D graphics are plotted with the help of software package.The unified solver technique extract vital solutions in explicit way. It is an easy-to-use method applied to obtain various exact solutions of nonlinear partial differential equations arising in fluid mechanics, nuclear, plasma and particle physics.

Wavelets approach for the solution of nonlinear variable delay differential equations

Abstract In this study, the Laguerre wavelet-oriented numerical scheme for nonlinear first and second-order delay differential equations (DDEs) is offered. The proposed technique is dependent on the truncated series of the Laguerre wavelets approximation of an unknown function. Here, we transform the different ordered DDEs into a system of non-linear algebraic equations with the help of limit points of a sequence of collocation points. Four nonlinear illustrations are involved to prove the efficiency of the planned technique. the Obtained results are equated with the current results, indicating the proposed technique’s accuracy and efficiency.

Plenteous specific analytical solutions for new extended deoxyribonucleic acid (DNA) model arising in mathematical biology

In this paper, the generalized Kudryashov (GK) approach and the sine-Gordon expansion approach are used for constructing new specific analytical solutions of the deoxyribonucleic acid model, which include the well-known bell-shaped soliton, kink, singular kink, periodic soliton, contracted bell-shaped soliton and anti-bell-shaped soliton. The efficacy of these strategies demonstrates their utility and efficiency in addressing a wide range of integer and fractional-order nonlinear evolution problems. The physical relevance of the demonstrated results has been proven using three-dimensional forms. It is interesting to mention that the solutions achieved here using the provided methods are extra-extensive and may be used to explain the internal interaction of the deoxyribonucleic acid model originating in mathematical biology. The suggested approach was utilized to get exact traveling wave solutions for fractional nonlinear partial differential equations appearing in nonlinear science.

Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency.

Transcendental liouvillian solutions of first order nonlinear differential equations

Transcendental liouvillian solutions of first order nonlinear differential equations

Numerical Modeling of Lyapunov Exponents for Structural Damage Identification

The main purpose of this article is to discuss the use of the Lyapunov exponents to evaluate the integrity of structures. The use of such coefficients is examined in an analysis that considers the geometric and physical nonlinearities, aiming to ensure the applicability of the method in robust simulations. The material nonlinearity is modeled using the multilinear isotropic elastoplastic model together with a recently developed damage model. The nonlinear equilibrium equations solution is obtained using the positional finite element method. The Newmark time-marching procedure is implemented to evaluate the Lyapunov coefficients and a nonlinear predictor technique that needs a single data series is employed. A numerical example of a frame structure is presented to illustrate the methodology applicability. Its results show that the Lyapunov exponents can be used as indicative parameters of structural integrity, since its analysis was able to detect the occurrence of the destabilization of the structure with the dynamic jump and the presence of material failures. The non-linear predictor proved to be an efficient technique for obtaining the Lyapunov exponents, with a low computational cost. The methodology presented to monitor structural integrity was shown to be a promising alternative.

An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

In this paper, we propose an effective numerical method using two-dimensional Shifted fractional-order Gegenbauer Multi-wavelets to find the approximate solutions of the time-fractional distributed order non-linear partial differential equations. The method is applied to numerically solve the fractional distributed order non-linear Klein–Gordon equation. We derive an exact formula for the Riemann-Liouville fractional integral operator for the Shifted fractional Gegenbauer Multi-wavelets. Applying function approximations obtained by this method turns the considered equation into a system of algebraic equations. Error estimation and convergence analysis of the method are also studied. Some numerical examples are included to show and check the effectiveness of the proposed method.

Existence and convergence of the least energy sign-changing solutions for nonlinear Kirchhoff equations on locally finite graphs

In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − ( a + b ∫ V | ∇ u | 2 d μ ) Δ u + c ( x ) u = f ( u ) on a locally finite graph G = ( V , E ), where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution u b of the above equation if c ( x ) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c ( x ) u = f ( u ).

A Homotopy-Based Approach to Solve the Power Flow Problem in Islanded Microgrid with Droop-Controlled Distributed Generation Units

This paper proposes a homotopy-based approach to solve the power flow problem (PFP) in islanded microgrid networks with droop-controlled distributed generation (DG) units. The technique is based on modifying an “easy” problem solution that evolves with the computation of intermediate results to the PFP solution of interest. These intermediate results require the solution of nonlinear equations through Newton–Raphson (NR) method. In favor of convergence, the intermediate solutions are close to each other, strengthening the convergence qualities of the technique for the solution of interest. The DG units are modeled with operational power limits and three types of droop-control strategies, while the loads are both magnitude voltage- and frequency-dependent. To evaluate the method performance, simulations are performed considering the proposed and classical NR methods, both departing from a flat start estimation. Tests are carried out in three test systems. Different load and DG unit scenarios are implemented for a 6-, 38-, and 69-bus test system. A base case is studied for all systems, while for the two larger models, a loading factor is used to simulate the load augmenting up to the maximum value. The results demonstrated that for the largest-size model system, only the homotopy-based approach could solve the PFP for stringent requirements such as the diversification of the load profile and hard loading operation point.

On decaying and asymptotically constant solutions of nonlinear equations with the Weyl fractional derivative of an order in [formula omitted

We consider a sublinear fractional differential equation of an order in the interval (1,2) where the fractional derivative is of the Weyl type. Existence and asymptotic behavior of decaying and asymptotically constant positive solutions is studied. We mainly deal with regularly varying coefficients and/or solutions, but we also allow a more general setting. Our results are sharp and in the special case where the coefficient in the equation is asymptotically equivalent to a power function and the order of the equation is 2 we get back known results. An important role in the proofs is played by the fractional Karamata integration theorem and other properties of regularly varying functions, fixed point principle, and generalized fractional L’Hospital rule.

A sufficient condition for global existence of the solution to nonlinear damped wave equations at arbitrary positive initial energy

AbstractWe consider the global existence of the solution for nonlinear damped wave equations at arbitrary positive initial energy, that is, . It is well known from a work of G. Filippo and S. Marco [Ann. Inst. H. Poincaré Anal. Non Linéaire, 2006], that which initial data with high energy make the model exists global solutions is still an interesting question. Here, we give it an affirmative answer. For this purpose, we suggest a unified approach, which is also valid where various forms of damping are present. Applying it, we show a sufficient condition to get the global existence of the solution for problem nonlinear damped wave equations.

Number of positive periodic solutions for feedback-driven nonlinear differential equation: application to hematopoietic process

Number of positive periodic solutions for feedback-driven nonlinear differential equation: application to hematopoietic process

Nonrelativistic limit and nonexistence of stationary solutions of nonlinear Dirac equations

In this paper, we study the nonrelativistic limit of the following nonlinear Dirac equations−icα⋅∇u+mc2βu−ωu+V(x)u=W(x)|u|p−2u, where c denotes the speed of light, m>0 is the mass of the electron. Then we show nonexistence of special solutions by using the nonrelativistic limit of the corresponding nonlinear Dirac equations. Here, nonexistence means no nontrivial solutions.

Stochastic solutions and singular partial differential equations

Stochastic solutions is a robust technique previously used to obtain new exact solutions for deterministic nonlinear partial differential equations as well as numerical algorithms suited to parallel computing. Here it is proposed as a solution method for partial differential equations driven by distribution-valued noises. Two examples are worked out in detail.

Equivalence of "generalized" solutions for nonlinear parabolic equations with variable exponents and diffuse measure data

We prove the equivalence of suitably defined weak solutions of a nonhomogeneous initial-boundary value problem for a class of nonlinear parabolic equations. We also develop the notion of both "renormalized" and "entropy" solutions with respect to the "generalized" $p(\cdot)$-capacity, initial datum, and diffuse measure data (which does not charge the set of null $p(\cdot)$-capacity). Conditions, under which "generalized weak" solutions of the nonhomogeneous problem are in fact well-defined, are also given.

Nonlinear optics problem with transformation of a spatial variable and an oblique derivative

In this paper, we consider a functional differential equation of parabolic type on a strip with a transformation of a spatial variable and boundary conditions with an oblique derivative. Using the Laplace and Fourier transforms, we obtain a representation of the problem under consideration in the form of a nonlinear integral equation. A special case of this representation is considered. The proved statements make it possible to implement iterative methods for obtaining approximate solutions of nonlinear partial differential equations taking into account the given conditions. The results show that the presented method is promising for solving similar problems.

Common fixed point result for multivalued mappings with applications to systems of integral and functional equations

In this paper, we establish a common fixed point theorem for two multivalued mappings satisfying some dominated conditions on a complete metric space. This new rational type contractive inequality refines various results in the literature. The main theorem is illustrated with examples. As application, we found the existence of solutions of system nonlinear integral equations and functional equations.

Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2

This article is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where \(\delta,\eta\in\mathbb{C}\) and \(g(z_1,z_2)\) is a polynomial in \(\mathbb{C}^2\). Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of tr

Symmetries and exact solution of certain nonlinear fractional ordinary differential equations

Symmetries and exact solution of certain nonlinear fractional ordinary differential equations