Order Nonlinear Differential Equations Research Articles

Exploring the magnetohydrodynamic stretched flow of Williamson Maxwell nanofluid through porous matrix over a permeated sheet with bioconvection and activation energy

The evolution of compact density heat gadgets demands effective thermal transportation. The notion of nanofluid plays active role for this requirements. A comparative account for Maxwell nanofluids and Williamson nanofluid is analyzed. The bioconvection of self motive microorganisms, non Fourier heat flux and activation energy are new aspects of this study. This article elaborates the effects of viscous dissipation, Cattaneo–Christov diffusion for Maxwell and Williamson nanofluid transportation that occurs due to porous stretching sheet. The higher order non-linear partial differential equations are solved by using similarity transformations and a new set of ordinary differential equations is formed. For numerical purpose, Runge–Kutta method with shooting technique is applied. Matlab plateform is used for computational procedure. The graphs for various profiles .i.e. velocity, temperature, concentration and concentration of motile micro-organisms are revealed for specific non-dimensional parameters. It is observed that enhancing the magnetic parameter M, the velocity of fluid decreases but opposite behavior happens for temperature, concentration and motile density profile. Also the motile density profile decrease down for Pe and Lb. The skin friction coefficient is enhanced for both the Williamson and Maxwell fluid.

On the behaviour of solutions to a kind of third order nonlinear neutral differential equation with delay

This paper presents a novel class of third order nonlinear nonautonomous neutral differential equation with delay. The third order neutral differential equation is cut down to a system of first order, a suitable complete Lyapunov-Krasovskii's functional is constructed and used, to obtain standard conditions on the nonlinear functions to ensure stability and uniform asymptotic stability of the trivial solutions, the existence of a unique periodic solution, uniform boundedness and uniform ultimate boundedness of solutions when the forcing term is nonzero. The obtained results are new and include many prominent results on neutral and nonneutral delay differential equations in literature. Finally, the practicability and reliability of the theoretical results are demonstrated.

Stagnation point flow of a viscous incompressible fluid

A steady two-dimensional incompressible viscous fluid flow in the vicinity of a stagnation point has been discussed. The reduced third order nonlinear ordinary differential equation of velocity has been solved using fourth order Runge-Kutta method with shooting technique. The qualitative (graphical) as well as quantitative (tabular) analysis of the velocity distribution has been made. Some of the important findings are (i) The viscous effect are confined in the boundary layer for fluids with very small viscosity. (ii) Outside the boundary layer, the viscous effect is negligible and the fluid is considered as inviscid. (iii) The difference in velocity between adjacent layers is decreasing for less viscous fluids.

Solving a class of variable order nonlinear fractional integral differential equations by using reproducing kernel function

<abstract><p>In this paper, reproducing kernel interpolation collocation method is explored for nonlinear fractional integral differential equations with Caputo variable order. In order to testify the feasibility of this method, several examples are studied from the different values of parameters. In addition, the influence of the parameters of the Jacobi polynomial on the numerical results is studied. Our results reveal that the present method is effective and provide highly precise numerical solutions for solving such fractional integral differential equations.</p></abstract>

On the distribution of zeros of all solutions of a first order nonlinear neutral differential equation

We investigate the distribution of zeros of all solutions of a non-autonomous nonlinear neutral differential equation that generalizes a lossless transmission network model. The neutral term is taken to be positive. We give several new estimations of the gap between adjacent zeros. The obtained results are supported by illustrative examples.

KAZE-SAR: SAR Image Registration Using KAZE Detector and Modified SURF Descriptor for Tackling Speckle Noise

Synthetic aperture radar (SAR) image registration is still a challenging task in remote sensing. The scale-invariant feature transform (SIFT) method and its extensions are most widely used feature detectors in SAR images. However, due to the presence of nonlinear speckle noise, some wrong features are chosen which directly influence the feature matching process. In this article, for the first time, the KAZE algorithm with a modified version of speeded-up robust features (SURF) descriptor is used for SAR image registration. It uses discrete stochastic second order nonlinear partial differential equations (PDEs) to model the SAR image’s edge structure. KAZE uses nonlinear diffusion filtering to build up the scale levels of the SIFT descriptor. It preserves the edges while simultaneously decreasing the speckle noise in smoothing the image. Therefore, it captures the exact location of features in down-sampling and filtering stages of SIFT. Experimental results show that our proposed method has increased the localization accuracy and distinctiveness of feature extraction through the use of KAZE detector. Our method is also able to handle challenging cases in different image sources, different view angles, different image times, complex affine mapping, presence of blurring, and noise.

The existence of the bounded solutions of a second order nonhomogeneous nonlinear differential equation

In this paper, we consider a second order nonlinear differential equation and establish two new theorems about the existence of the bounded solutions of a second order nonlinear differential equation. In these theorems, we use different Lyapunov functions with different conditions but we get the same result. In addition, two examples are given to support our results with some figures.

RETRACTED: A novel approach for EMHD Williamson nanofluid over nonlinear sheet with double stratification and Ohmic dissipation

The current article highlights the non-Newtonian Williamson nanofluid with electro-magnetohydrodynamic (EMHD) flow over a nonlinear expanding sheet. Thermal and solutal stratification effects are considered due to the higher temperature difference and the impact of variable viscosity along with Ohmic dissipation is also incorporated. Transformation is applied for the conversion of physical partial differential equations (PDEs) into non-dimensional higher order nonlinear ordinary differential equations (ODEs). A well-known analytical approach known as the homotopy analysis method (HAM) is effectively applied to solve the differential equations. Different non-dimensional emerging parameters such as Weissenberg and Hartman number, Brownian motion and stratification parameters, stretching index, viscosity parameter, and Lewis number are used to check their impacts on velocity, concentration, and temperature profiles. To acquire the optimal solution through HAM, [Formula: see text] -curves are drawn. In the tabulated form, the numerical values for the non-dimensional Nusselt number and skin friction are arranged.

Boubaker Wavelets Functions: Properties and Applications

تم في هذا البحث تقديم شرح تفصيلي لدوال متعددة حدود بوبكر المتعامدة مع بعض الخواص ذات الاهمية، كذلك استنتاج تعريف متعددات حدود بوبكر الموجية في الفترة (1, 0] وذلك بالاستفادة من بعض الخواص المهمة لمتعددة حدود بوبكر. تمتلك هذه الدوال الاساسية خاصية العيارية المتعامدة بالإضافة الى ضرورة تواجد المنطلق المرصوص. لهذه الدوال الموجية العديد من المزايا وقد استخدمت في المجال النظري بالإضافة الى المجال العملي وتم استخدامها مع متعددات الحدود المتعامدة لغرض طرح طريقة جديدة للتعامل مع العديد من المسائل في العلوم والهندسة ولذلك تعتبر طريقة استخدام الموجبات ذات اهمية كبيرة عند الاستفادة منها في المجالات ذات العلاقة. بالإضافة الى الاستفادة من موجبات بوبكر للحصول على خاصية جديدة وهي مشتقات دالة بوبكر الموجية. استخدمت موجية بوبكر مع طريقة الترصيف للحصول على حل عددي تقريبي لمعادلات لان ايمدن من النوع الخطي المنفرد. تصف معادلات لان ايمدن العديد من الظواهر المهمة في علم الرياضيات والفيزياء السماوي مثل الانفجارات الحرارية الكونية وتكوين النجوم. وتعتبر احدى حالات مسائل القيمة الابتدائية المنفردة للمعادلات التفاضلية اللاخطية من الرتبة الثانية. تقوم هذه الطريقة المقترحة بتحويل معادلة لان ايمدن الى نظام من المعادلات التفاضلية الخطية والتي يمكن حلها بسهولة باستخدام الحاسبة. بناءً على هذا فقد ظهر تطابق الحل العددي مع الحل التحليلي بالرغم من استخدام عدد قليل من متعددات حدود بوبكر الموجية لغرض ايجاد هذا الحل. كذلك، تم في هذا البحث البرهنة على قيمه قيد الخطأ المستخرج من هذه الطريقة. وتضمن هذا البحث على ثلاث امثلة عددية من نوع معادلات لان ايمدن لتوضيح قابلية استخدام الطريقة المقترحة. تم توضيح النتائج الحقيقة مع النتائج التقريبية في شكل جداول ورسوم هندسية لغرض المقارنة.

On Lyapunov-type inequalities for (n+ 1)st order nonlinear differential equations with the antiperiodic boundary conditions

In this paper, we establish new Lyapunov-type inequalities for $\left(n+1\right)$st order nonlinear differential equation including $p$-relativistic operator and $q$-prescribed curvature operator under the antiperiodic boundary conditions.

The existence of solutions for Sturm\u2013Liouville differential equation with random impulses and boundary value problems

In this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

Monotone solutions of first order nonlinear differential systems

This article concerns the classification, continuablity, boundedness, and existence of solutions for a system of first order nonlinear differential equations. First, we prove that all solutions of the system are eventually monotonic and can be separated into two classes. Then we discuss the continuability of solutions. After that we establish necessary and sufficient conditions for the boundedness of all solutions. Also, we study the existence of monotone solutions in certain classes.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/93/abstr.html

Monotonic Solutions of Second Order Nonlinear Difference Equations

Classification, boundedness, and existence of solutions of a second order nonlinear difference equation are investigated. First, it is proved that all solutions are eventually monotone. Then, the necessary and sufficient conditions for the boundedness of all solutions are established. Finally, the existence of different types of monotonic solutions are presented. The obtained results have extended and improved some existing ones.

Existence of periodic solutions of second-order nonlinear random impulsive differential equations via topological degree theory

In this paper, we investigate the existence of periodic solutions for a class of second order nonlinear random impulse differential equations. By extending the definitions of continuous function bound set, curvature bound set and Nagumo set in topological degree to PC space, and defining the appropriate operators and conditions, using the theory of topological degree and coincidence degree, we prove that the solution must be bounded if it exists and find the boundary value. Finally, we define appropriate Nagumo set and autonomous curvature bound set, and obtain the existence of periodic solutions of equations and simultaneous equations by using Mawhin’s continuity theorem.

Oscillation and nonoscillation criteria for second order difference equations with generalized difference operators

In this study we investigate some new oscillation and nonoscillation criteria and generalize and improve some results in the literatures for second order nonlinear difference equation with generalized difference operators of the form ∆l,a(pn∆l,axn) + qn(∆l,axn) β = F (n, xn, ∆l,bxn), where ∆l,σ is generalized difference operator such that defined as ∆l,σxn = xn+l − σxn, and F : N × R 2→ R˙ . Also, some examples illustrating the results are

Numerical approximations for a class of nonlinear higher order singular boundary value problem by using homotopy perturbation and variational iteration method

In this work, we focus on the non-linear fourth order class of singular boundary value problem which contains the parameter λ . We convert this non-linear differential equation into third order non-linear differential equation. The third order problem is singular, non self adjoint, and nonlinear. Moreover, depending upon λ , it admits multiple solutions. Hence, it is too difficult to capture these solutions by any discrete method such as finite difference and so forth. Here we develop an iterative technique with the help of homotopy perturbation method and variational iteration method in a suitable way. Convergence of this series solution is studied in a novel way. We compute these solutions numerically. For small positive values of λ , singular BVP has two solutions while solutions can not be found for large positive values of λ . Furthermore, we also find dual solutions for λ < 0 .

Numerical approach for enhanced mass transfer of Bio-convection on Magneto-hydrodynamic Carreau fluid flow through a nonlinear stretching surface

The present study deals with an investigation of steady two dimensional flow of Carreau fluid covering both the motile microorganism and nanoparticles through a nonlinear stretching surface. The idea of microorganisms is implied just to stabilize suspended nanoparticles by convection. The governing PDEs (partial differentiable equations) are transform set of nonlinear ODEs (ordinary differentiable equations) using suitable similarity transformation. The higher order nonlinear differential equation is solved by using Runge-Kutta-Felhberg method with shooting technique. The performance of fluid microorganism density, concentration, temperature, velocity in different values of physical parameters are scrutinized. Here the numerical results are presented through graphs and tables. Numerical values of skin friction, local Nusselt, Sherwood and local density numbers are analyzed. This study is relevant to navel microorganisms using few technologies combining with the Nano fluid with Bio-convection.

Explicit positive solutions to G-heat equations and the application to G-capacities

In this paper, we study a class of explicit positive solutions to G -heat equations by solving second order nonlinear ordinary differential equations. Based on the positive solutions, we give the sharp order of G -capacity c σ ( [ − ε , ε ] ) when ε → 0 .

Reduction of systems of two nonlinear parabolic type partial differential equations to solvable forms using differential invariants

Differential invariants for linear and nonlinear ordinary and partial differential equations have been derived using Lie infinitesimal method. These invariants help in reducing differential equations to their simplest possible solvable forms through invertible transformations of the dependent and independent variables. Here we derive differential invariants for a class of systems of two second order nonlinear parabolic type partial differential equations. Deduced invariants are shown to reveal solvable forms of these systems of PDEs that are much simpler than the considered general systems of nonlinear parabolic type PDEs.

Three Dimensional MHD Hybrid Nanofluid Flow with Rotating Stretching/Shrinking Sheet and Joule Heating

A three-dimensional hybrid nanofluid flow over a stretching/shrinking sheet is numerically studied. The hybrid nanofluid being considered in this study used water as the base fluid and mixed with two types of solid nanoparticles, namely alumina (Al2O3) and copper (Cu). The main focus of the current study is to examine the effect of magnetic field, Joule heating, and rotating sheet on the velocity, and temperature profiles. In addition, the impact of suction and stretching sheet on the variations of reduced skin friction, , and reduced heat transfer are studied as well. The fluid flow and heat transfer problem presented in this study is governed by a system of nonlinear partial differential equations (PDEs), which is then transformed into the corresponding system of high order nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting system of higher order nonlinear ODEs is solved numerically using a boundary value solver known as bvp4c, which operates on the MATLAB computational platform. Results revealed that dual solutions exist for shrinking sheet while unique solutions are observed for stretching sheet with various values of Cu nanoparticles volume fraction and magnetic parameter. Dual solutions also exist for the value of the suction parameter greater than its critical point with various values of Cu nanoparticles volume fraction. Velocity profile of the hybrid nanofluid increases alongside with the value of magnetic parameter but declination was observed in the profile of and temperature, for both solutions as the value of Cu nanoparticles volume fraction increases. When the value of rotational parameter increases, both velocity and profiles increase for both solutions. This indicates that the momentum boundary layer thickness increases with increasing values of for both solutions, but thermal boundary layer thickness decreases for the first solution and increases for the second solution. Finally, an increment in the value of Eckert number causes the temperature of the hybrid nanofluid to rise as well for both first and second solutions.