On Global Dynamics in Duffing Equation with Quasiperiodic Perturbation

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

Non-linear parameter estimation in pesticide degradation

The aim of this study is to provide numerical solutions to magnetohydrodynamic (MHD) cross flow with heat and mass transfer over a stretching sheet in a hybrid nanofluid with suction/injection and convective boundary conditions. This will be accomplished by presenting numeric solutions. Nanoparticles of graphene oxide and silver were suspended in kerosene oil as the base fluid taken into consideration is a nanofluid hybrid. After that, the physical flow problem is modeled as a set of partial differential equations, which are then transformed into an equivalent system of nonlinear ordinary differential equations (ODEs) by making use of the relevant similarity transformations. To gain different graphical and numerical results for analyzing the influence of numerous physical restrictions on velocity, heat, and mass profiles, the system of ODEs is solved using the computing power of the Lobatto IIIA technique. The plot of the velocity profile, temperature, concentration, entropy generation and Bejan number with separating magnitude like that power index number ( 0.1 < n < 2.0 0.1\lt n\lt 2.0 ), Weissenberg number ( 0.1 < We < 1.0 0.1\lt {\rm{We}}\lt 1.0 ), Suction/injection ( 0.1 < S < 1.0 0.1\lt S\lt 1.0 ), Magnetic parameter (0.1 < M < 1.0), Biot number ( 0.1 < Bi < 0.3 0.1\lt {\rm{Bi}}\lt 0.3 ), Radiation parameter ( 0.1 < Rd < 2.0 0.1\lt {\rm{Rd}}\lt 2.0 ), Schmidt number ( 1.1 < Sc < 2.0 1.1\lt {\rm{Sc}}\lt 2.0 ), and Chemical reaction parameter ( 1.1 < Sc < 2.0 1.1\lt {\rm{Sc}}\lt 2.0 ). We noted that when the Weissenberg parameter ( We {\rm{We}} ) increases, the flow speed will increase as well. Also, the increasing values of M M slows the flow speed but the opposite effect is observed in the temperature profile. Moreover, the greater values of M M , We, and S reduced in the local skin friction. It is concluded that hybrid nanofluid is better than ordinary nanofluid by percentage values. The results are used in industry and technology in fluid-based systems with shrinkable or flexible materials, MHD flow meters, pumps, and the drawing of plastic strips. Because hybrid nanoparticles speed up the cooling process, they are important in cutting and production.

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations.

A stochastic computing approach is implemented in the present work to solve the nonlinear nanofluidics system that occurs in the model of atomic physics. The process converts the partial differential nanofluidics system with suitable level of similarities transformation into nonlinear systems of differential equations. For the construction of datasets, finite difference scheme (Lobatto IIIA) is applied through different selection of collocation points for nonlinear nanofluidics system having accuracy of order four. Lobatto IIIA has a strong point to tackle extremely nonlinear systems of ordinary differential equations in smooth way. For different scenarios, datasets are well trained through computing scheme to investigate the heat transfer and thermal performance of nanofluidic transportation system of nanofluids and hybrid nanofluids toward stretching surfaces with variation of Biot number, Nusselt number and skin fraction. Furthermore, the reliability, accuracy and efficiency are endorsed through various statistical analysis and graphical illustrations of proposed computing scheme.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

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.

<div class="section abstract"><div class="htmlview paragraph">In this paper, a new solution method is presented to study the effect of wave propagation in engine manifolds, which includes solving one-dimensional models for compressible flow of air. Velocity, pressure, and density profiles are found by solving a system of non-linear Partial Differential Equations (PDEs) in space and time derived from Euler’s equations. The 1D model includes frictional losses, area change, and heat transfer. The solution is traditionally found by utilizing the Method of Characteristics and applying finite difference solutions to the resulting system of ordinary differential equations (ODEs) over a discretized grid. In this work, orthogonal collocation is used to solve the system of ODEs that is defined along the characteristic curves. Orthogonal polynomials are utilized to approximate velocity, pressure, sound speed, and the characteristic curves along which the system of PDEs reduce to a system of ODEs. The approximation polynomials are defined over the whole manifold domain, transforming Euler’s equations into a system of ODEs that can be solved using a generic ODE solver. This reduction is done symbolically using a computer algebra system (Maple). The method results in a system of ODEs that has a higher spatial order along the whole space compared to methods based on finite differences, reducing the number of nodes required to find an acceptable solution that captures the state dynamics at different locations inside the manifold in real time. The proposed model is compared against the Method of Characteristics (MOC) that is used as a reference model; this comparison includes the states at the inlet, outlet, and midpoint. In summary, a high order method that can calculate solutions of the 1D manifold model equations is developed by finding the respective polynomial approximations along the 1D space and solving the resultant system using a generic solver in real time.</div></div>

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.

An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations

Heat transfer of Maxwell base fluid flow of nanomaterial with MHD over a vertical moving surface

Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations

Exterior panels forming the exterior skin of flight vehicles traveling through the atmosphere at high supersonic speeds are often susceptible to the occurrence of limit-cycle type self-excited vibrations called flutter. Panel flutter is resulting from the dynamic instability of the aerodynamic, inertia, and elastic forces of the system. The equations of motion for panel flutter are derived using Von Karman's large deflection plate theory and quasi-steady aerodynamic theory. The equations are given in terms of the displacement and are presented in non-dimensional form. Galerkins method is used to transform the system of nonlinear partial differential equations into a system of nonlinear ordinary differential equations in time variables. The obtained equations are then solved by the method of harmonic balance and Newton-Raplison algorithm. The results are compared with both theoretical and experimental results given in publications. Then, a parametric study is performed to study the effects of aerodynamic loading, aerodynamic damping, structural damping, in-plane applied loads, thermal stresses and cavity pressure.

The numerous industrial and engineering applications of Casson nanofluid is due to the superiority of its thermophysical properties. Tomatoes paste, engine oil, soup etc., are examples of Casson fluid and when nanometer-sized particles are suspended in such Casson fluid, it becomes Casson nanofluid. This paper considers a natural convective magnetohydrodynamics flow of Cu-engine oil nanofluid across a convectively heated vertical plate. The effects of self-heating of the fluid (measured by the Eckert number), internal conductive resistance to external convective resistance (measured by the Biot number), magnetic field strength, volume fraction of the nanoparticles on the temperature and velocity of mass and heat transfer of Casson nanofluid is analysed. An appropriate model governing the flow of Casson nanofluid is formulated as a system of nonlinear partial differential equations. The natural convection boundary condition is included. To solve the problem, an appropriate similarity transformation is used to reformulate the system as a system of nonlinear ordinary differential equations. The shooting technique is used to convert the boundary problem to initial value problems before Runge-Kutta method, with the Gills constants, is used to solve the reformulated problem. The results are depicted as graphs. Flow velocity is found to increase as the base fluid becomes more Casson and as nanoparticle volume fraction increases. It is also found that increasing Eckert number, Biot number and magnetic field strength causes an increase in the flow temperature.