Order Nonlinear Equations Research Articles

Application of Fixed Point Theorem to Solvability for Non-Linear Fractional Hadamard Functional Integral Equations

In the present paper, our main work aims to discover the existence result of the fractional order non-linear Hadamard functional integral equations on [1,a] by employing the theory of measure of non-compactness together with the fixed point theory in Banach space. An example is presented to see the validity and practicability of our existence result.

New oscillation criteria for third order nonlinear mixed neutral dynamic equations

New oscillation criteria for third order nonlinear mixed neutral dynamic equations

Inelastic soliton wave solutions with different geometrical structures to fractional order nonlinear evolution equations

The general time fractional Burger- Fisher (TF-BF) and the space–time regularized long-wave (STF-RLW) equations are considered as examples of gravitational water waves in cold plasma as well as so many areas. The above equations are used in nonlinear science and engineering to study long waves in seas and harbors that travel in just one direction. First, the two equations are transformed to ODEs by applying a fractional complex transform along with characteristics of confirmable fractional derivative (CFD). Then, the extended tanh-function (ET-F) approach is investigated to find a variety of analytical solutions with different geometrical wave structures the mentioned models. The results are in the form of kink, one-, two-, multiple-solitons solutions, and other types sketched in 2D, 3D, and contour patterns.

Nonlinear dynamical modelling of high frequency electrostatic drift waves using fluid theoretical approach in magnetized plasma

A novel third order nonlinear evolution equation governing the dynamics of the recently observed high frequency electrostatic drift waves has been derived in the framework of a plasma fluid model in an inhomogeneous magnetized plasma. The corresponding linear dispersion relation arising out of the fluid equations has been studied in detail for the conventional low frequency as well as the recently observed high frequency electrostatic drift waves along with the electrostatic ion cyclotron waves. The derived nonlinear evolution equation of third order is then decomposed into two second order equations as the order of the equation becomes reduced after this kind of decomposition under certain conditions, and one equation becomes linear whereas the other equation becomes nonlinear. The detailed analysis of fixed points leading to bifurcations has been performed using the qualitative theory of differential equations and the bifurcation theory of planar dynamical systems for the reduced second order nonlinear equation. The bifurcation curves for the individual fixed points are also shown for the variations of different parameters in the parameter space of this reduced nonlinear equation. Then some exact as well as approximate travelling wave solutions of this reduced nonlinear equation have been derived. As the other second order reduced equation is linear in nature, its exact oscillatory and exponential solutions are obtained. The intersection of the solutions of these two reduced second order equations provides the solutions of the original third order nonlinear evolution equation; it is verified that the solutions of the reduced nonlinear second order equation are subsets of the oscillatory solution of the reduced linear second order equation in most cases whereas the intersection is different if the exponential solution of the reduced linear second order equation is considered. This further implies that the solutions of the reduced second order nonlinear equation can directly represent the solutions of the original third order nonlinear evolution equation in most cases representing the dynamics of the nonlinear high frequency electrostatic drift waves. These solutions of the novel third order nonlinear evolution equation represent certain vortex-like structures as our work is restricted to (1 + 1) dimensions only. The possible applications of our novel results on the dynamics of the high frequency electrostatic drift waves are discussed along with future directions for research in this field.

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

Method of own forms for determination of separate bands

The problem of determining the separate bands of the phosphor photoluminescence spectrum described by the Gaussian function is solved by the method of decomposition into its own forms. This allows you to calculate two parameters of each band: amplitude and variance. An example is the photoluminescence spectrum of a ZnS:Mn single crystal. It consists of five separate bands. Two systems of fifth-order equations linear for one parameter of each band are obtained. This parameter is generalized and is equal to the product of the amplitude and the root of the variance. The coefficients of these systems depend on the second parameter. This non-linear parameter is related to variance. In order to simplify the two systems, the problem is solved in several stages. Based on the analysis of literary sources, the two largest bands are selected. In addition, the origin of coordinates is well chosen. This allows us to simplify both systems and completely determine the parameters of each of the largest bands from the systems of second-order equations. As a result of these simplifications, the order of the two systems and the parameters of the other three bands are reduced. Thus, the problem is reduced to solving a system of the second and third order nonlinear equations.

A Quantitative Approach to $$n{\text {th}}$$-Order Nonlinear Fuzzy Integro-Differential Equation

A Quantitative Approach to $$n{\text {th}}$$-Order Nonlinear Fuzzy Integro-Differential Equation

Bound/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations

We construct two classes of time discretization schemes for fourth order nonlinear equations by combining a function transform approach with the scalar auxiliary variable (SAV)approach. A suitable function transform ensures that the schemes are bound/positivity preserving, while the SAV approach enables us to construct unconditionally stable schemes. The first class of schemes requires solving a coupled second-order linear systems with variable coefficients at each time step, while the second class of schemes only requires solving a fourth-order linear equation with constant coeffcients. We apply this approach to the Cahn-Hilliard equations with logarithmic potential and the more challenging Lubrication-type equations, and present ample numerical examples to validate the accuracy and efficiency of the proposed schemes.

A New and Simple Mathematical Technique to Study the Steady-state Performance of Isolated Asynchronous Generator

Isolated Asynchronous Generator (IAG) is nowadays widely used for renew-able power generation from the sources like wind and small hydro. Tradi-tionally, the steady-state analysis of IAG is carried out by solving a complexhigher order non-linear polynomial equation obtained in a complicated wayfrom the steady-state circuit diagram. In this paper, a simple new mathe-matical procedure has been introduced to obtain two non-linear polynomialequations in much more simplified form which can easily be solved forthe unknown variables i.e., per unit generated frequency (a) and magnetiz-ing reactance (Xm). Between these two equations, one equation comprisesonly one unknown variable‘a’ and hence, easy to solve. Differential SearchAlgorithm (DSA) has been efficiently implemented for solving these non-linear equations. The computational efficacy of DSA has been compared withthat of Newton-Raphson (N-R) method, Linear Search Algorithm (LSA),Binary Search Algorithm (BSA) and Particle Swarm Optimization (PSO) technique. The performance of the IAG has been studied under differentoperating conditions such as variation of speed, capacitance and load. Allthe simulated results have been experimentally verified using a three-phase,415 volts, 2.2 kW, star-connected induction generator and a close agreementhas been found.

Quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters

This paper is concerned with construction of quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters. The boson–fermion correspondence for these symmetric functions have been presented. In virtue of quantum fields, we derive a series of infinite order nonlinear integrable equations, namely, universal character hierarchy, symplectic KP hierarchy and symplectic universal character hierarchy, respectively. In addition, the solutions of these integrable systems have been discussed.

Oscillation of even order nonlinear dynamic equations on time-scales

In this paper, the authors discuss the oscillatory behavior of solutions to a class of even order nonlinear dynamic equations on time scales. The results are established by a comparison with n-th order delay dynamic inequalities or first-order delay dynamic equations whose oscillatory characters are known. Several corollaries are obtained for special cases.

Lyapunov-type inequalities for third order nonlinear equations

Lyapunov-type inequalities for third order nonlinear equations

Integrable classes of a family of evolution equations

The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained, and the results are transformed into known integrable equations from the literature. Recursion operators are also given for all the symmetry integrable classes found, some of which are the only ones known for some of the canonical classes.

The approximate analytical solutions of nonlinear gas dynamics equation within caputo-fabrizo fractional operator

The adomian decomposition method (ADM) is applied to solve fractional order nonlinear gas dynamic equation with Caputo-Fabrizo fractional operator in this paper. In this method, illustration example evaluate the accuracy and applicability of the mentioned procedure. The outcomes got by ADM are in acceptable concurrence with the special arrangement of the problem.

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

In this paper, we establish some new oscillation criteria for second- order nonlinear neutral delay dynamic equation of the form (r(t)((m(t)y(t) ? p(t)y(τ(t)))∆)γ )∆ + ƒ (t, y(?(t))) = 0 on a time scale

A Family of Exponential Attractors and Inertial Manifolds for a Class of Higher Order Kirchhoff Equations

In this paper, the global dynamics of a class of higher order nonlinear Kirchhoff equations under n-dimensional conditions is studied. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup associated with the initial boundary value problem are proved, and the existence of a family of exponential attractors is obtained. Then, by constructing the corresponding graph norm, the condition of a spectral interval is established when N is sufficiently large. Finally, the existence of the family of inertial manifolds is obtained.

Bound/Positivity Preserving and Unconditionally Stable Schemes for a Class of Fourth Order Nonlinear Equations

Bound/Positivity Preserving and Unconditionally Stable Schemes for a Class of Fourth Order Nonlinear Equations

Asymptotic properties of solutions to discrete Volterra type equations

We study higher order nonlinear discrete Volterra equations of the form We discuss the problem of the existence of solutions with prescribed asymptotic behavior. In the proof of the main theorem we use the iterated remainder operator and the Schauder's type fixed point theorem. This approach allows us to control the degree of approximation. Our results are also new for the special cases of the above equation such as linear Volterra equation, ordinary difference equation and discrete Emden–Fowler type equation.

Oscillation of nonlinear neutral dynamic equations on time scales

The authors present necessary and sufficient conditions for the oscillation of a class of second order non-linear neutral dynamic equations with non-positive neutral coefficients by using Krasnosel’skii’s fixed point theorem on time scales. The nonlinear function may be strongly sublinear or strongly superlinear.

Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials

We establish the boundedness properties of the Neumann extension operator in half-space in the setting of Morrey–Lorentz spaces. As a by-product we derive estimates on the restriction operator in block spaces. Direct application to solvability of a fourth order nonlinear equation related to higher order boundary conformally invariant problem is considered. By employing a nonvariational approach, we obtain a unique solution under suitable smallness conditions on the boundary data and potentials prescribed in Morrey–Lorentz spaces which allow for singular functions. Moreover, these solutions are shown to be C∞ in the interior, Cloc2,μ(R+n+1¯), μ∈(0,1) in some special case and satisfy interesting qualitative properties including positivity. The results are extended to a larger class of problems involving the polyharmonic operator. In particular, the higher order nonlocal equation (−Δ)2m−12v=K(x)|v|σ−1v+H(x)v+ginRn,m∈Nfor 2≤2m<n+1 is solvable in a suitable Morrey–Lorentz space and admits positive solutions whenever σ>n/(n−2m+1). We also prove that no positive solution in the latter framework exists if σ≤n/(n−2m+1).