Solutions Of Nonlinear Differential Equations Research Articles

Approximate Solution of Nonlinear Differential Equations with the Help of Rational Spline Functions

Approximate Solution of Nonlinear Differential Equations with the Help of Rational Spline Functions

Painlevé analysis and traveling wave solutions of the sixth order differential equation with non-local nonlinearity

In this paper, we study a nonlinear partial differential equation for describing high dispersion optical soliton with non-local nonlinearity. Taking into account the traveling wave reduction, we get system of ordinary differential equations (ODEs) for real and imaginary parts of the original equation. To determine the integrability of equation we apply the Painlevé test for analysis of obtained ODE system. We illustrate that the system of equations does not have the Painlevé property since there is only one integer Fuchs index. However using the Painlevé data we find the compatibility conditions for the ODE system. Under these conditions, the traveling wave solution of nonlinear differential equations are constructed and illustrated.

Existence and stability of solution for nonlinear differential equations with [formula omitted]-Hilfer fractional derivative

In this paper, we investigate a nonlinear ψ-Hilfer fractional integro-differential equation on an unbounded domain [a,+∞). Firstly, the existence and uniqueness of solution for the equation are proved by means of Banach contraction mapping principle based on suitable growth conditions in an appropriate Banach space. Moreover the stability of Ulam–Hyers–Rassias, Ulam–Hyers and Semi-Ulam–Hyers–Rassias to the initial value problem is obtained.

Entire solutions of certain nonlinear differential and delay-differential equations

Let f(z) be an entire solution of the nonlinear differential equation f(z)n+P(z,f)=b1eλ1z+b2eλ2z, where n≥2, P(z,f) is a differential polynomial in f of degree ≤n−1 with meromorphic functions of order less than 1 as coefficients, and b1, b2, λ1, λ2 are nonzero constants and t=λ2/λ1 is real. By utilizing Nevanlinna theory, we show that t=−1 or t is a positive rational number and that in either case f(z) is expressed in terms of exponential functions. This partially answers a question in the literature. We also show that these results extend to finite-order entire solutions of the above equation when P(z,f) is replaced by a delay-differential polynomial in f with meromorphic functions of order less than 1 as coefficients.

Kannan Contraction Operator on the Domain of r . -Cesàro Matrix in ℓ t . and Related Prequasi Ideal with an Application of Nonlinear Difference Equations

In this article, the sequence space Ξ r , t υ has been built by the domain of r l -Cesàro matrix in Nakano sequence space ℓ t l , where t = t l and r = r l are sequences of positive reals with 1 ≤ t l < ∞ , and υ f = ∑ l = 0 ∞ ∑ z = 0 l r z f z / ∑ z = 0 l r z t l , with f = f z ∈ Ξ r , t . Some topological and geometric behavior of Ξ r , t υ , the multiplication maps acting on Ξ r , t υ , and the eigenvalues distribution of operator ideal constructed by Ξ r , t υ and s -numbers have been examined. The existence of a fixed point of Kannan prequasi norm contraction mapping on this sequence space and on its prequasi operator ideal are investigated. Moreover, we indicate our results by some explanative examples and actions to the existence of solutions of nonlinear difference equations.

Meromorphic Solutions of Certain Nonlinear Difference Equations

Let $$p_1, p_2$$ be nonzero rational functions, $$\alpha _1, \alpha _2$$ be nonzero constants, and $$P_d(z, f)$$ be a difference polynomial in f of degree d. We prove that every finite order meromorphic solution of nonlinear difference equations $$f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}$$ has few poles provided $$n\ge 2$$ and $$d\le n-1$$ . This result shows there exists some difference between the existence of finite order meromorphic solutions of the above equations and its corresponding differential equations. We also give the forms of finite order meromorphic solutions of the above equations under some conditions on $$\alpha _1/\alpha _2$$ .

Travelling Wave solutions of hyperbolic Telegraph equation by Tanh method

Tanh method is utilized to find travelling solutions of second order nonlinear Telegraph equation. As a result, we attain dissimilar travelling wave solutions. Our aim is to show that this method is most efficient and convenient approach for verdict travelling wave solutions of nonlinear differential equations. For calculation the software MAPLE is used.

Solitary waves of the generalized Sasa–Satsuma equation with arbitrary refractive index

Traveling wave reduction of the generalized Sasa–Satsuma equation with arbitrary refractive index is considered. The system of equations corresponding to the real and imaginary parts is presented. The two compatibility conditions to this system equations are given. Methods for finding exact solutions of nonlinear differential equations with arbitrary powers of nonlinearity are presented and discussed. Periodic and solitary wave solutions of the generalized Sasa–Satsuma equation with arbitrary refractive index are found.

Modified iteration method for numerical solution of nonlinear differential equations arising in science and engineering

In this paper, a modified iteration method (MIM) has been proposed to solve nonlinear second-order ODEs. Convergence analysis and error estimate of the proposed method are also discussed. Computational efficiency of this method is illustrated through numerical examples.

Solving Riccati-Type Nonlinear Differential Equations with Novel Artificial Neural Networks

In this study we investigate neural network solutions to nonlinear differential equations of Ricatti-type. We employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but avoid the standard back-propagation algorithm for updating the intrinsic weights. Our objective is to minimize an error, which is a function of the network parameters i.e., the weights and biases. Once the weights of the neural network are obtained by our systematic procedure, we need not adjust all the parameters in the network, as postulated by many researchers before us, in order to achieve convergence. We only need to fine-tune our biases which are fixed to lie in a certain given range, and convergence to a solution with an acceptable minimum error is achieved. This greatly reduces the computational complexity of the given problem. We provide two important ODE examples, the first is a Ricatti type differential equation to which the procedure is applied, and this gave us perfect agreement with the exact solution. The second example however provided us with only an acceptable approximation to the exact solution. Our novel artificial neural networks procedure has demonstrated quite clearly the function approximation capabilities of ANN in the solution of nonlinear differential equations of Ricatti type.

SOLVABILITY FOR A COUPLED SYSTEM OF PERTURBED IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH PERIODIC AND ANTI-PERIODIC BOUNDARY CONDITIONS

In this paper, we first provide a refinement result for the abstract continuation theorem for $k$-set contractions. The new version of the theorem is equivalent to the usual one and it better adapts to study the existence of solutions for nonlinear differential equations. Then we discuss a new class of coupled system of implicit fractional boundary value problem. The nonlinear terms of equations involving perturbations, and the boundary conditions are constituted by periodic and anti-periodic boundary conditions. Based on the abstract continuation theorem for $k$-set contractions, an interesting existence result is obtained. Finally, an example is constructed for illustrating the application of our main results.

Thermodynamic Consistency and Thermomechanical Dynamics (TMD) for Nonequilibrium Irreversible Mechanism of Heat Engines

The irreversible mechanism of heat engines is studied in terms of thermodynamic consistency and thermomechanical dynamics (TMD) which is proposed for a method to study nonequilibrium irreversible thermodynamic systems. As an example, a water drinking bird (DB) known as one of the heat engines is specifically examined. The DB system suffices a rigorous experimental device for the theory of nonequilibrium irreversible thermodynamics. The DB nonlinear equation of motion proves explicitly that nonlinear differential equations with time-dependent coefficients must be classified as independent equations different from those of constant coefficients. The solutions of nonlinear differential equations with time-dependent coefficients can express emergent phenomena: nonequilibrium irreversible states. The couplings among mechanics, thermodynamics and time-evolution to nonequilibrium irreversible state are defined when the internal energy, thermodynamic work, temperature and entropy are integrated as a spontaneous thermodynamic process in the DB system. The physical meanings of the time-dependent entropy, T(t)dS(t), , internal energy, dƐ(t), and thermodynamic work, dW(t), are defined by the progress of time-dependent Gibbs relation to thermodynamic equilibrium. The thermomechanical dynamics (TMD) approach constitutes a method for the nonequilibrium irreversible thermodynamics and transport processes.

Дослідження динаміки завантаження вібраційних машин з вертикальним збуренням

The purpose of research. The main purpose of the work is to develop mathematical software for studying the dynamics of vibration machines of volumetric processing, and calculate the influence of various factors on the efficiency of the process based on applied systems of automated mathematical calculations, including MathCad and MatLab. Method. The research was carried out on the basis of a class of vibrating machines with unbalanced drive type and spring suspension. A mathematical model of loading vibrating processing machines with vertical perturbation is constructed, which is represented by a layering of flat beams that perform vertical oscillations. Methods of nonlinear mechanics were used to build a mathematical model. Results. The dynamic processes at vibration compaction of loading and separation of its fractions for the purpose of increase of intensity of the given technological processes are investigated. The dependences for determining the influence of physical and mechanical properties of loading components on the process dynamics are obtained. Scientific novelty. New approaches for construction of mathematical models of research of processes in machines of vibration processing, in particular vibration consolidation and separation are considered. In the course of research, differential equations were used to determine the change in the amplitude and frequency of loading from the influence of external and internal parameters of the vibration processing process. Practical significance. The obtained mathematical model makes it possible to investigate the dynamics of loading of the vibrating machine, to determine the amplitude and frequency of oscillations of loading of the vibrating machine depending on its properties and the properties of the vibrating machine. The obtained solutions of nonlinear differential equations to describe the motion of loading in vibrating machines allow to automatically determine the amplitudes, frequencies and trajectories of different components of loading depending on the parameters of the system "vibrating machine - loading", by algorithmizing them in applied systems of automated mathematical calculations.

Spherical Symmetric Solitons of Interacting Spinor, Scalar and Gravitational Fields in General Relativity

The concept of soliton as regular localized stable solutions of nonlinear differential equations is being widely utilized in pure science for various aims. In present analysis, the soliton concept is used as a model in order to describe the configurations of elementary particles in general relativity. To this end, our study deals with the spherical symmetric solitons of interacting Spinor, Scalar and Gravitational Fields in General Relativity. Thus, exact spherical symmetric general solutions to the interaction of spinor, scalar and gravitational field equations have been obtained. The Einstein equations have been transformed into a Liouville equation type and solved. Let us emphasize that these solutions are regular with localized energy density and finite total energy. In addition, the total charge and spin are limited. Moreover, the obtained solutions are soliton-like solutions. These solutions can be used in order to describe the configurations of elementary particles.

Integrability via Functional Expansion for the KMN Model

This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.

Thermo diffusion aspects in Jeffrey nanofluid over periodically moving surface with time dependent thermal conductivity

Double diffusion flow of Jeffrey fluid in presence of nanoparticles is studied theoretically under time dependent thermal conductivity. The considered nanoparticles are evaporated over convectively heated surface which moves periodically in its own plane. The appropriate dimensionless variables are employed to obtain the dimensionless forms of governing equations. We computed the analytical solution of non-linear differential equations by utilizing homotopy analysis method. The present investigation reveals the features of various emerging parameters like Deborah number, combined parameter, oscillation frequency to stretching rate ratio, Prandtl number, Lewis number, thermophoresis parameter, Brownian motion parameter, nano Lewis number, modified Dufour parameter, and Dufour solutal Lewis number. A useful enhancement in movement of nanoparticles is observed by utilizing the combined magnetic and porosity effects. Unlike traditional studies, present analysis is confined with the unsteady transportation phenomenon from periodically moving surfaces. Such computation may be attributable in flow results from tensional vibrations due to stretching and elastic surfaces. The simulation presented here can be attractable significance in the bioengineered nanoparticles manufacturing. It is observed that the heat transportation of nanoparticles may efficiently enhance through the utilization of variable thermal conductivity. The solutal concentration decreases with increasing Deborah number and Lewis number. It is further noted that the nano Lewis number causes reduction of nanoparticles concentration.

Construction of exact traveling wave solutions of the Bogoyavlenskii equation by [formula omitted]-expansion and [formula omitted]-expansion techniques

In this article, we construct exact solutions of the Bogoyavlenskii equation using (1/G′)-expansion and (G′/G,1/G)-expansion techniques. Both techniques have been successfully implemented to obtain exact solutions including hyperbolic, complex trigonometric, trigonometric and rational solutions of the Bogoyavlenskii equation. 3D, contour and 2D graphics are presented of the solutions obtained for different special values. Further, the advantages and disadvantages of both the techniques have been discussed in this study. The proposed techniques are reliable and applicable for attaining wave solutions of nonlinear differential equations. Also, these techniques can greatly minimize the size of computing work compared to other available techniques.

A new multistage technique for approximate analytical solution of nonlinear differential equations

The article introduces a new multistage technique for solving a polynomial system of nonlinear initial and boundary value problems of differential equations. The radius of convergence R of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Then guided by the convergence-control parameter h<R, the domain of the problem is split into subintervals. By stepping out in a multistage manner, corresponding subproblems are defined which are then subsequently solved with conventional Parker-Sochacki method to get a piecewise continuous solution with very high accuracy. The method is applied to SIR epidemic model, stiff differential equation modelling combustion, Lorenz chaotic problem, and the Troesch's boundary value problem. The results obtained showed a remarkable accuracy when compared with Runge-Kutta Method of order 4. The article showcased the proposed method as a simple, yet accurate approximate analytical technique for nonlinear differential equations.

Singular solutions of nonlinear differential equations with p(t)-Laplacian

This paper deals with the singular solutions of the nonlinear differential equation (a(t)|x′|p(t)−2x′)′=b(t)|x|q(t)−2x involving p(t)-Laplacian. Sufficient conditions are given for the existence and the nonexistence of singular solutions of the first (second) kind. In addition, we study the existence of proper solutions and the coexistence of proper solutions and singular solutions of the second kind. Finally, some examples are given to illustrate our results.

Positive periodic solution to an indefinite singular equation

In this paper, we prove the existence of a positive periodic solution for nonlinear differential equation with an indefinite singularity by using Green’s function and fixed point theorem in cones. The results are applicable to weak as well as strong singularities. At last, we give one example to show the application of the theorem.