Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

Efficient Numerical Solution of the Modified Mild-Slope Equation

An efficient numerical model for wave refraction,diffraction and reflection is presented in thispaper.In the model,the modified time-dependent mild-slope equation is transformed into an evolutionequation and an improved ADI method involving a relaxation factor is adopted to solve it.The methodhas the advantage of improving the numerical stability and convergence rate by properly determining therelaxation factor.The range of the relaxation factor making the differential scheme unconditionally stableis determined by stability analysis.Several verifications are performed to examine the accuracy of the pres-ent model.The numerical results coincide with the analytic solutions or experimental data very well,andthe computer time is reduced.

Numerical solutions to heat transfer of nanofluid flow over stretching sheet subjected to variations of nanoparticle volume fraction and wall temperature

The numerical analysis of heat transfer of laminar nanofluid flow over a flat stretching sheet is presented. Two sets of boundary conditions(BCs) are analyzed, i.e., a constant(Case 1) and a linear streamwise variation of nanoparticle volume fraction and wall temperature(Case 2). The governing equations and BCs are reduced to a set of nonlinear ordinary diferential equations(ODEs) and the corresponding BCs, respectively. The dependencies of solutions on Prandtl number P r, Lewis number Le, Brownian motion number N b, and thermophoresis number N t are studied in detail. The results show that the reduced Nusselt number and the reduced Sherwood number increase for the BCs of Case 2 compared with Case 1. The increases of N b, N t, and Le numbers cause a decrease of the reduced Nusselt number, while the reduced Sherwood number increases with the increase of N b and Le numbers. For low Prandtl numbers, an increase of N t number can cause to decrease in the reduced Sherwood number, while it increases for high Prandtl numbers.

Numerical Solution of Constrained Mechanical System Motions Equations and Inverse Problems of Dynamics

In this paper the method of design of kinematical and dynamical equations of mechanical systems, applied to numerical ealization, is proposed. The corresponding difference equations, which are obtained, give a guarantee of computations with a given precision. The equations of programmed constraints and those of constraint perturbations are defined. The stability of the programmed manifold for numerical solutions of the kinematical and dynamical equations is obtained by corresponding construction of the constraint perturbation equations. The dynamical equations of system with programmed constraints are set up in the form of Lagrange’s equations in generalized coordinates. Certain inverse problems of rigid body dynamics are examined.

Existence and Numerical Solution for a Coupled System of Multi-term Fractional Differential Equations

An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.

Numerical Solution of the Weight Function for Electromagnetic Flowmeter

A numerical solution of the weight function for a two-electrode electromagnetic flowmeter was proposed. The solution was obtained by using the finite element method based on the basic equation of a traditional two-electrode electromagnetic flowmeter. The two-dimensional distribution of the weight function of the electromagnetic flowmeter obtained was verified by the analytical solution. Three-dimensional distribution of the weight function was also presented in the paper. It can be employed to analyze the sensitivity and linearity of the electromagnetic flowmeter with non-uniform magnetic field,and even to assist the design of the excitation coil pair.

Numerical solution of a mathematical model for water wavesin large coastal areas

Based on the evolution equation for water waves, a mathematical model for wave propaga tion in large mild - slope areas is derived. The model is solved by the finite difference method with the staggered grid system. The computational results are in good agreement with experimental data and show that the model can obtain better results with relatively coarser grids. The model can be used to simulate water wave propagation in large coastal areas and can be efficiently solved without much programming effort.

Numerical Solution of Euler-Lagrange Equation with Caputo Derivatives

In this paper the fractional Euler-Lagrange equation is considered.The fractional equation with the left and right Caputo derivatives of order a∈(0,1]is transformed into its corresponding integral form.Next,we present a numerical solution of the integral form of the considered equation.On the basis of numerical results,the convergence of the proposed method is determined.Examples of numerical solutions of this equation are shown in the final part of this paper.

Rheological Analysis of CNT Suspended Nanofluid with Variable Viscosity: Numerical Solution

In this article, we discuss the two-dimensional stagnation-point flow of carbon nanotubes towards a stretching sheet with water as the base fluid under the influence temperature dependent viscosity. Similarity transformations are used to simplify the governing boundary layer equations for nanofluid. This is the first article on the stagnation point flow of CNTs over a stretching sheet with variable viscosity. A well known Reynold's model of viscosity is used. Single wall CNTs are used with water as a base fluid. The resulting nonlinear coupled equations with the relevant boundary conditions are solved numerically using shooting method. The influence of the flow parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are explored and presented in forms of graphs and interpreted physically.

MODELLING AND NUMERICAL SOLUTIONS OF A GAUGE PERIODIC TIME DEPENDENT GINZBURG-LANDAU MODEL FOR TYPE-Ⅱ SUPERCONDUCTORS

In this paper we seek the solutions of the time dependent Ginzburg-Landau model for type-Ⅱ superconductors such that the associated physical observables are spatially periodic with respect to some lattice whose basic lattice cell is not necessarily rectangular. After appropriately foring the gange, the model can be formulated as a system of nonlinear parabolic partial differential equations with quasi-periodic boundary conditions. We first give some results concerning the existence, uniqueness and regularity of solutions and then we propose a semiimplicit finite element scheme solving the system of nonlinear partial dmerential equations and show the optimal error estimates both in the L2 and energy norm.We also report on some numerical results at the end of the paper.

A Study of Traveling Wave Structures and Numerical Investigation of Two-Dimensional Riemann Problems with Their Stability and Accuracy

The Riemann wave system has a fundamental role in describing waves in various nonlinear natural phenomena,for instance,tsunamis in the oceans.This paper focuses on executing the generalized exponential rational function approach and some numerical methods to obtain a distinct range of traveling wave structures and numerical results of the two-dimensional Riemann problems.The stability of obtained traveling wave solutions is analyzed by satisfying the constraint conditions of the Hamiltonian system.Numerical simulations are investigated via the finite difference method to verify the accuracy of the obtained results.To extract the approximation solutions to the underlying problem,some ODE solvers in FORTRAN software are applied,and outcomes are shown graphically.The stability and accuracy of the numerical schemes using Fourier’s stabilitymethod and error analysis,respectively,to increase the reassurance are investigated.A comparison between the analytical and numerical results is obtained and graphically provided.The proposed methods are effective and practical to be applied for solving more partial differential equations(PDEs).

New Soliton Wave Solutions to a Nonlinear Equation Arising in Plasma Physics

The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics,physics,and engineering disciplines.This article intends to analyze several traveling wave solutions for themodified regularized long-wave(MRLW)equation using several approaches,namely,the generalized algebraic method,the Jacobian elliptic functions technique,and the improved Q-expansion strategy.We successfully obtain analytical solutions consisting of rational,trigonometric,and hyperbolic structures.The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation.The adaptive moving mesh method evenly distributes the points on the high error areas.This method perfectly and strongly reduces the error.We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used.To better understand the considered equation’s physical meaning,we present some 2D and 3D figures.The exact and numerical approaches are efficient,powerful,and versatile for establishing novel bright,dark,bell-kink-type,and periodic traveling wave solutions for nonlinear PDEs.

Numerical Simulation of Diffusion Type Traffic Flow Model Using Second-Order Lax-Wendroff Scheme Based on Exponential Velocity Density Function

In order to control traffic congestion, many mathematical models have been used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation, which provides a non-linear second-order parabolic partial differential equation. The analytical solution of the diffusion-type traffic flow model is very complicated to approximate the initial density of the Cauchy problem as a function of x from given data and it may cause a huge error. For the complexity of the analytical solution, the numerical solution is performed by implementing an explicit upwind, explicitly centered, and second-order Lax-Wendroff scheme for the numerical solution. From the comparison of relative error among these three schemes, it is observed that Lax-Wendroff scheme gives less error than the explicit upwind and explicit centered difference scheme. The numerical, analytical analysis and comparative result discussion bring out the fact that the Lax-Wendroff scheme with exponential velocity-density relation of diffusion type traffic flow model is suitable for the congested area and shows a better fit in traffic-congested regions.

A Novel Method for Linear Systems of Fractional Ordinary Differential Equations with Applications to Time-Fractional PDEs

This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering and science.An approximate solution of the system is sought in the formof the finite series over the Müntz polynomials.By using the collocation procedure in the time interval,one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure.This technique also serves as the basis for solving the time-fractional partial differential equations(PDEs).The modified radial basis functions are used for spatial approximation of the solution.The collocation in the solution domain transforms the equation into a system of fractional ordinary differential equations similar to the one mentioned above.Several examples have verified the performance of the proposed novel technique with high accuracy and efficiency.

A Galerkin-Type Fractional Approach for Solutions of Bagley-Torvik Equations

In this study,we present a numerical scheme similar to the Galerkin method in order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2.In this approach,the approximate solution is assumed to have the form of a polynomial in the variable t=xα,whereαis a positive real parameter of our choice.The problem is firstly expressed in vectoral form via substituting the matrix counterparts of the terms present in the equation.After taking inner product of this vector with nonnegative integer powers of t up to a selected positive parameter N,a set of linear algebraic equations is obtained.After incorporation of the boundary conditions,the approximate solution of the problem is then computed from the solution of this linear system.The present method is illustrated with two examples.

Numerical Analysis of Laterally Loaded Long Piles in Cohesionless Soil

The capability of piles to withstand horizontal loads is a major design issue.The current research work aims to investigate numerically the responses of laterally loaded piles at working load employing the concept of a beam-on-Winkler-foundation model.The governing differential equation for a laterally loaded pile on elastic subgrade is derived.Based on LegendreGalerkin method and Runge-Kutta formulas of order four and five,the flexural equation of long piles embedded in homogeneous sandy soils with modulus of subgrade reaction linearly variable with depth is solved for both free-and fixed-headed piles.Mathematica,as one of the world’s leading computational software,was employed for the implementation of solutions.The proposed numerical techniques provide the responses for the entire pile length under the applied lateral load.The utilized numerical approaches are validated against experimental and analytical results of previously published works showing a more accurate estimation of the response of laterally loaded piles.Therefore,the proposed approaches can maintain both mathematical simplicity and comparable accuracy with the experimental results.

Analysis of Convective Transport of Temperature-Dependent Viscosity forNon-Newtonian Erying Powell Fluid: A Numerical Approach

Non-Newtonian is a type of fluid that does not comply with the viscosity under the Law of Newton and is being widely used in industrial applications.These include those related to chemical industries,cosmetics manufacturing,pharmaceutical field,food processing,as well as oil and gas activities.The inability of the conventional equations of Navier–Stokes to accurately depict rheological behavior for certain fluids led to an emergence study for non-Newtonian fluids’models.In line with this,a mathematical model of forced convective flow on non-Newtonian Eyring Powell fluid under temperature-dependent viscosity(TDV)circumstance is formulated.The fluid model is embedded with the Newtonian heating(NH)boundary condition as a heating circumstance and is assumed to move over a stretching sheet acting vertically.Using appropriate similarity variables,the respective model was converted into ordinary differential equations(ODE),which was later solved utilizing the Keller box approach.The present model is validated by comparing the existing output in literature at certain special limiting cases,where the validation results display a firm agreement.The current outputs for the proposed model are shown in tabular and graphical form for variation of skin friction plus Nusselt number,velocity and temperature distribution,respectively.

Numerical Study of Fluid Flow over Bundle Tubes

In this paper, a bank of tubes containing a flowing fluid which is immersed in a cross flow second medium of fluid with different temperature has been studied numerically using computational fluid dynamics. Laminar steady flow with a low Reynolds number has been studied in this work. Inlet mass flow rate and the bulk temperature are known and numerical method has been implemented to study the convective heat transfer to investigate the temperature and flow fields. Effects of different inlet bulk temperatures and mass flow rates have been investigated on temperature and pressure variations.

AN ANALYTICAL SOLUTION TO LARGE DEFLECTION EQUATIONS OF SIMPLY－SUPPORTED RECTANGULAR HYPERBOLOIDAL SHALLOW SHELLS OF ORTHOTROPIC COMPOSITES

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