A Metaheuristic Optimization Algorithm for Solving Higher-Order Boundary Value Problems

An effective metaheuristic algorithm to solve the higher-order boundary value problems, called a genetic programming technique is presented. In this paper, a genetic programming algorithm, which depends on the syntax tree representation, is employed to obtain the analytical solutions of higher- order differential equations with the boundary conditions. The proposed algorithm can be produce an exact or approximate solution when the classical methods lead to unsatisfactory results. To illustrate the efficiency and accuracy of the designed algorithm, several examples are tested. Finally, the obtained results are compared with the existing methods such as the homotopy analysis method, the B-Spline collocation method and the differential transform method.

An analytical approach of heat transfer modelling with thermal stresses in circular plate by means of Gaussian heat source and stress function

In the proposed research work we have used the Gaussian circular heat source. This heat source is applied with the heat flux boundary condition along the thickness of a circular plate with a nite radius. The research work also deals with the formulation of unsteady-state heat conduction problems along with homogeneous initial and non-homogeneous boundary condition around the temperature distribution in the circular plate. The mathematical model of thermoelasticity with the determination of thermal stresses and displacement has been studied in the present work. The new analytical method, Reduced Differential Transform has been used to obtain the solution. The numerical results are shown graphically with the help of mathematical software SCILAB and results are carried out for the material copper.

Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method

This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also compute analytical or semi-analytical results to the proposed problems. Also by some proper examples we demonstrate the results.

Comparison Between Numerical Methods for Generalized Zakharov system

We present two numerical methods to get approximate solutions for generalized Zakharov system GZS. The first one is Legendre collocation method, which assumes an expansion in a series of Legendre polynomials , for the function and its derivatives occurring in the GZS, the expansion coefficients are then determined by reducing the problem to a system of algebraic equations. The second is differential transform method DTM , it is a transformation technique based on the Taylor series expansion. In this method, certain transformation rules are applied to transform the problem into a set of algebraic equations and the solution of these algebraic equations gives the desired solution of the problem.The obtained numerical solutions compared with corresponding analytical solutions.The results show that the proposed method has high accuracy for solving the GZS.

Casson fluid flow with heat and mass transfer in a channel using the differential transform method

In the present investigation, we consider the heat and mass transfer characteristics of steady, incompressible and electrically conducting Casson fluid flow in a channel. The effect of chemical reactions have also been considered. The differential transform method (DTM) is applied to a system of non-linear ODEs, and the results are obtained in the form of DTM series. The principal gain of this approach is that it applies to the non-linear ODEs without requiring any discretization, linearization or perturbation. The velocity, mass and heat transfer profiles thus obtained are in good agreement with those provided by the quasi-linearization method (QLM). Graphical results for velocity, concentration and temperature fields are presented for a certain range of values of the governing parameters.

SPLINES SOLUTIONS OF HIGHER-ORDER BVPs THAT ARISE IN CONSISTENT MAGNETIZED FORCE FIELD

In this paper, cubic polynomial and nonpolynomial splines are developed to solve solutions of 10th- and 12th-order nonlinear boundary value problems (BVPs). Such types of BVPs occur when a consistent magnetized force field is applied crosswise the fluid in the substance of gravitational force. We will amend our problem into such a form that converts the system of [Formula: see text]th- [Formula: see text] [Formula: see text]th-order BVPs into a new system of [Formula: see text]nd-order BVPs. The appropriate outcomes by using CP Spline and CNP Spline are compared with the exact root. To show the efficiency of our results, absolute errors calculated by using CP Spline and CNP Spline have been compared with other methods like differential transform method, Adomian decomposition method, variational iteration method, cubic B-spline, homotopy perturbation method, [Formula: see text]th- and [Formula: see text]th-order B-spline and our results are very encouraging. Graphs and tables are also presented in the numerical section of this paper.

Analysis of arrhenius activation energy and chemical reaction in nanofluid flow and heat transfer Over a thin moving needle

Objective: A numerical and theoretical study is developed to analyze the combined effect of activation energy and chemical reaction in the flow of nanofluids due to the thin moving needle using the mathematical nanofluid model offered by Buongiorno. A passively controlled nanoparticle volume fraction boundary is assumed rather than actively controlled. Methods: A similarity transformation is utilized to convert the governing partial differential equations to a set of ordinary differential equations which are then solved numerically by Runge-Kutta Shooting Method (RKSM). The physical characteristics of flow, heat and mass transfer are illustrated via graphs and tables for some set of values of governing parameters. Results: In addition, the basic non-linear governing equations are solved analytically using semi-analytical technique called Differential transform method (DTM) and the comparison has been made with the numerical and the published results. Conclusion: The present study reveals that the ratio between the needle velocity and the composite velocity brings out to increase the velocity distribution with λ<0. Moreover, the activation energy influences the chemical species to react from the thickness of the concentration layer η=0.6 and the fraction of nanoparticles to the fluid is significantly more away from the needle surface.

Bioconvection of nanofluid flow in a thin moving needle in the presence of activation energy with surface temperature boundary conditions

A comparative analysis is formed to analyze the combined effects of a binary chemical reaction and activation energy in the flow of bio-nanofluid due to the thin moving needle using the mathematical nanofluid model offered by Buongiorno with different boundary conditions namely, Newtonian heating and prescribed surface temperature. The governing partial differential equations converted into a set of final controlled governing physical flow equations by using similarity variables and then solved numerically by Runge–Kutta–Fehlberg method along with shooting technique and analytically by differential transform method. The results gained for the dimensionless velocity, temperature, concentration, motile diffusivity number, Nusselt number and Sherwood number are presented through graphs and tables. The present study reveals that an enhancement in the pertinent parameters has considerably altered the physical characteristics of the flow and heat transfer, which applies in biosensors and biomedical instrumentation. Also, the rate of heat transfer from the needle to the fluid is controlled by applying Newtonian heating than by applying prescribed surface temperature against all the parameters. In addition, we carried out the statistical analysis to determine the dependence of the physical parameter on the rate of heat transfer for both cases of heating process.

Solving a Higher-Dimensional Time-Fractional Diffusion Equation via the Fractional Reduced Differential Transform Method

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions.

H-U-Type Stability and Numerical Solutions for a Nonlinear Model of the Coupled Systems of Navier BVPs via the Generalized Differential Transform Method

This paper is devoted to generalizing the standard system of Navier boundary value problems to a fractional system of coupled sequential Navier boundary value problems by using terms of the Caputo derivatives. In other words, for the first time, we design a multi-term fractional coupled system of Navier equations under the fractional boundary conditions. The existence theory is studied regarding solutions of the given coupled sequential Navier boundary problems via the Krasnoselskii’s fixed-point theorem on two nonlinear operators. Moreover, the Banach contraction principle is applied to investigate the uniqueness of solution. We then focus on the Hyers–Ulam-type stability of its solution. Furthermore, the approximate solutions of the proposed coupled fractional sequential Navier system are obtained via the generalized differential transform method. Lastly, the results of this research are supported by giving simulated examples.