Order Nonlinear Equations Research Articles

On existence and numerical solution of higher order nonlinear integro-differential equations involving variable coefficients

We investigate the approximate solution to a class of fourth-order Volterra–Fredholm integro differential equations (VFIDEs). Additionally, we are able to get some adequate results for the existence of a solution with the use of nonlinear analysis techniques. The basis for the required numerical computation is provided by the Haar wavelet collocations (HWCs) technique, which converts the problems into a system of algebraic equations. Then, in order to obtain the required numerical solution, we solve the resulting system of algebraic equations using Gauss elimination and Broyden’s techniques. This method can be used to show convergence, and the predicted rate of convergence is two. Along with relevant examples, we have included a graphical presentation to demonstrate our suggested approach.

Numerical Solution of Two-Dimensional Fractional Order Diffusion Equation by using Elzaki Transform with Residual Power Series Method

The Elzaki transform with RPS method is an efficient and reliable approach for solution of non-integer order linear as well as non-linear differential equations. The major endeavourof current work is to find the solution of two dimensional non-integer order diffusion equation by Elzaki transform with RPS method. Firstly, Elzaki is applied on this equation. Secondly, inverse Elzaki is taken on this equation for finding the expressionof solution. Thirdly, assumed approximate solution is substituted on this equation. Then unidentified coefficient functions are obtained by using residual function is equal to zero as well as combining its initial circumstances. At last, coefficient functions are substituted in power series form for finding finite approximate logical solutions. The judgment between exact solution and approximate analytic solutions with different number of terms of this equation are determined and compared for reliability. This method reduces the size of computational works of solution of non-integer order non-linear diffusion equation

Topological Regularity for Solutions to the Generalised Hopf Equation

The generalised Hopf equation is the first order nonlinear equation defined on a planar domain Omega subset {mathbb {C}}, with data Phi a holomorphic function and eta ge 1 a positive weight on Omega , hwhw¯¯η(w)=Φ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} h_w\\,\\overline{h_{\\overline{w}}}\\,\\eta (w) = \\Phi . \\end{aligned}$$\\end{document}The Hopf equation is the special case eta (w)={tilde{eta }}(h(w)) and reflects that h is harmonic with respect to the conformal metric sqrt{{tilde{eta }}(z)}|dz|, usually eta is the hyperbolic metric. This article obtains conditions on the data to ensure that a solution is open and discrete. We also prove a strong uniqueness result.

Chaos control of fractional order nonlinear Bloch equation by utilizing sliding mode controller

Nuclear magnetic resonance is commonly used in engineering, chemistry, and medicine to study complex materials by relating magnetization to applied radiofrequency, gradient, and static magnetic fields through the Bloch equation. Chaos in the Bloch equation is a key factor in many important applications. This study focuses on the study of the Bloch equation under the influence of the Caputo fractional derivative, both with and without delay, and explores the underlying chaos using a sliding mode controller. The controller’s effectiveness is observed under uncertainty and external disturbances for both commensurate and incommensurate systems, and theoretical aspects such as the existence and uniqueness of solutions and the stability of the controlled system are examined. Lyapunov exponents are calculated for various fractional derivatives to demonstrate the presence of chaos in the system, and numerical simulations are used to verify theoretical assertions.

Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations

We investigate the tanh–coth method’s usefulness in understanding space–time fractional nonlinear evolution problems. To use this method and then examine their plots, we take into consideration three significant equations: the space–time fractional Burgers equation, the space–time fractional regularized long wave equation, and the space–time fractional Boussinesq equation. The modified Riemann–Liouville derivatives are used to define the fractional derivative. The approach provides a linearization- or small perturbation-free analytical solution in the form of a convergent series with easily calculable components. For the aforementioned nonlinear fractional equations, we have discovered their precise solutions. We utilize a generalized fractional complex transform to translate these to ordinary differential equations which subsequently resulted into number of exact solutions. The method provides us abundant analytical traveling wave solutions containing fewer number parameters with the aid of Mathematica and MATLAB.

Periodic travelling waves for a fourth order nonlinear evolution equation

In this article we provide a travelling wave analysis for a fourth order non linear evolution equation. In particular we prove the existence of periodic travelling waves while we exclude the existence of solitary waves for proper values of the parameters. Moreover, we analyse the set of stationary solutions and provide a new proof of the existence of limit cycle for a related equation, that is, the Van der Pol equation.

Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales

In this paper, we shall investigate the oscillatory properties of third order nonlinear delay dynamic equations. Applying suitable comparison theorems and by a Riccati transformation technique, we establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero. Our results not only unify the oscillation of third order nonlinear differential and difference equations but also can be applied to different types of time scales with

Analytical solutions of some fractional order nonlinear evolution equations by sine-cosine method

Analytical solutions of some fractional order nonlinear evolution equations by sine-cosine method

Analysis of new soliton type solutions to generalized extended (2 + 1)-dimensional Kadomtsev-Petviashvili equation via two techniques

In this paper, we examine the solitary wave solutions of the generalised (2 + 1) extended Kadomtsev–Petviashvili (eKP) equation, which is used as model for the surface waves and internal waves in straits or channels and describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. We secure distinct soliton solutions by employing two amelioration methodologies, namely the improved F-expansion method and the new Kudryashov method. These methods explore several unique forms of solitary wave solutions, including dark, rational, singular, periodic, and exponential ones. In order to illustrate propagation of some reported wave solutions, the graphs associated with these solutions are presented by choosing appropriate parametric values in 2D, 3D, contour, and density plots with the use of the symbolic software Mathematica 13.0. The presented solutions realize very significant fact for investigation the qualitative interpretations for various phenomena in our nature. The computed solutions revealed that the applied methods are influential, efficious and skilful and can be a best way to handle other higher order nonlinear evolution equations.

Bifurcation of traveling waves in a liquid film with broken time-reversal symmetry

According to the analysis of the lubrication theory, the surface wave will develop in the region of low Reynolds number, and the presence of an odd viscosity. Fourth order nonlinear wave equations (generalized Kuramoto-Sivashinsky equation) that appear in the theory of thin liquid films are introduced. Analyses of linear stability for sinusoidal waves in various regimes are investigated. The Hopf bifurcation points for the traveling waves are investigated. Surface wave stability will arise more quickly if an odd viscosity is present.

Quasilinear Fractional Order Equations and Fractional Powers of Sectorial Operators

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers Aγ for an operator A such that −A generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of Aγ for some γ∈(0,1) of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov–Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.

Regularity of Hamiltonian stationary equations in symplectic manifolds

In this paper, we prove that any C1-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond.

Lightcurve inversion problem for objects with negative Gaussian curvature

In this paper we consider a second order nonlinear hyperbolic equation of Monge-Ampère type arizing in the Light Curve Inversion problem [7,8,4] when dealing with objects with regions with negative gaussian curvature. The above works imply that such regions are of practical importance in the Light Curve Inversion problem. It is shown here that in such regions existence and uniqueness of solutions to the original equation can be obtained by solving a system of four first order quasi-linear differential equations supplemented by suitable initial and boundary conditions. Furthermore, these solutions can be found by integrating a system of ordinary differential equations. The numerical methods in [14] can be applied to actually find these solutions. We present here the completed part of the overall project, which establishes the necessary basis for efficient numerical solution. It is expected that the part of the project dealing with numerics will be addressed in a separate paper. This paper is written in a form accessible to physicists, engineers and applied mathematicians working on the Light Curve Inversion problem and design of reflectors.

A new 2-level compact off-step implicit method in exponential form for the solution of fourth order nonlinear parabolic equations

A new 2-level compact off-step implicit method in exponential form for the solution of fourth order nonlinear parabolic equations

Long-time behavior for fourth order nonlinear wave equations with dissipative and dispersive terms

The initial boundary value problem for a class of fourth-order nonlinear wave equations with dissipative terms and dispersive terms is considered in this paper. We first establish the global existence and the exponential decay of the solution. Next, based on Condition (C) and semigroup theory arguments, the existence of the global attractor is derived.

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

New soliton solutions and modulation instability analysis of fractional Huxley equation

In this research, the new auxiliary equation method (NAEM) for higher order nonlinear fractional Huxley equation is being employed to extricate the novel soliton solutions using Beta and M-Truncated fractional derivatives. For waves of finite amplitude, the Huxley equation demonstrates a substantial transfer of spectrum energy. A comparison of the solutions of the model with both fractional derivatives is also included in this research. Various kinds of solitary traveling wave solutions, such as trigonometric, hyperbolic, exponential, rational functions, etc., are found. These types of solutions demonstrate the superiority of the novelty of the method. This method’s key advantage over others is that it provides more broad solutions with certain flexible parameters. 3D and 2D graphs are used graphically to demonstrate the dynamical structures of the solutions. The results are presented in a way that demonstrate the usefulness and competence of the approach used to handle various nonlinear fractional partial differential equations. Lastly, we investigate the comparison of the gain spectra for modulation instability and the depiction of certain noteworthy outcomes by illustratively depicting the 2D figures produced by carefully considering the parameters

On the oscillations of the solutions for a class of second order nonlinear functional dynamic equations on time scales with unbounded neutral coefficients

On the oscillations of the solutions for a class of second order nonlinear functional dynamic equations on time scales with unbounded neutral coefficients

Oscillation of higher order nonlinear dynamic equations with a nonlinear neutral term

This study introduces the oscillation of higher order nonlinear dynamic equations with a nonlinear neutral term. The findings are obtained via utilising an integral criterion as well as a comparison theorem with the oscillatory properties of a first‐order dynamic equation. We propose novel oscillation criteria that improve, extend and simplify existing ones in the literature. The results are associated with several numerical examples.

A two-dimensional magnetohydrodynamic system: geometric decomposition and canonical reduction

A geometric decomposition previously adopted in areas of nonlinear continuum mechanics and soliton theory is applied to a two-dimensional time-independent magnetohydrodynamic system to obtain reduction to a canonical third order nonlinear equation subject to a constraint. Novel classes of exact solutions to the magnetohydrodynamic system are thereby constructed. Connection is made with kinematic conditions which attend certain steady two-dimensional motions of fibre-reinforced viscous fluids.