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# Numerical Methods: Problems And Solutions

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.

## Numerical methods: problems and solutions

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following:

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free-software alternative is the GNU Scientific Library.

Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis. Excel, for example, has hundreds of available functions, including for matrices, which may be used in conjunction with its built in "solver".

Algorithms for calculating the junction points between optimal nonsingular and singular subarcs of singular control problems are developed. The algorithms consist in formulating appropriate initialvalue and boundary-value problems; the boundary-value problems are solved with the method of multiple shooting. Two examples are detailed to illustrate the proposed numerical methods.

We try to compare the solutions by some numerical techniques when we apply the methods on some mathematical biology problems. The Runge-Kutta-Fehlberg (RKF) method is a promising method to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for insect population, one-species Lotka-Volterra model. The technique is described and illustrated by numerical examples. We modify the population models by taking the Holling type III functional response and intraspecific competition term and hence we solve it by this numerical technique and show that RKF method gives good results. We try to compare this method with the Laplace Adomian Decomposition Method (LADM) and with the exact solutions.

In the field of science and technology, numerous significant physical phenomena are frequently modeled by nonlinear differential equations. Such equations are often very difficult to solve analytically. Yet, analytical approximate methods are very important for obtaining the accurate solutions which have gained much significance in recent years. There are various methods, undertaken to find out approximate solutions to nonlinear problems. Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), Differential Transform Method (DTM), Variational Iteration Method (VIM), Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), and Runge-Kutta-Fehlberg (RKF) method are some very popular methods. The purpose of this paper is to bring out the numerical solution of various population models by using the approach, namely, Runge-Kutta-Fehlberg (RKF) method.

Recently, different scientists use the numerical method in their different problems. Here we try to give some references showing the importance of quasi-numerical methods techniques in present time: Kumar and Baskar  consider B-spline quasi-interpolation based numerical methods for some Sobolev type equations. Analytical and numerical study of dirty bosons in a quasi-one-dimensional harmonic trap is discussed by Khellil et al. . Numerical study of a quasi-hydrodynamic system of equations for flow computation at small Mach numbers is considered by Balashov and Savenkov . Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions is discussed by Liu et al. . Numerical solutions of quasi-two-dimensional models for laminar water hammer problems are considered by Zhao .

The accurate solutions of population growth models may become a difficult task if the equations are highly nonlinear. To overcome the situation there we take the numerical simulation, until there are no particular numerical methods for solving such problems. So to fill up the gap, here we find the approximated solutions of some population models by applying such some reliable, efficient, and more comfortable numerical technique (e.g., LADM, RKF) and try to conclude which one is the best.

The paper is organized as follows. The basic literature survey on population growth model, functional response, and numerical techniques is addressed in Section 1. In Section 2 we discuss the numerical methods, namely, Runge-Kutta-Felhberg (RKF) method and Laplace Adomian Decomposition Method (LADM) for nonlinear equation. The necessary algorithm for finding the numerical results is also discussed in this section. Section 3 is followed by a numerical example in different mathematical biology models. The different biological model is formulated and we write the general expression of the numerical solution. Section 4 is illustrated by the comparison study between the solutions and error terms of these said methods. Finally conclusions and future research scope of this paper are drawn in the last section, Section 5.

Remark 1. From Table 1 and Figure 1, we see the comparison among the RKF method, three-iterate LADM, and exact solutions for model I for particular numerical value of parameters and initial condition. The numerical results show that RKF method is of good accuracy.

Remark 2. In Table 2 and Figure 2, we show the comparison among the RKF method, three-iterate LADM, and exact solutions for model II in the case when , , , , , and . Again, the numerical results show that RKF method is of good accuracy.

In this paper, we describe the method for finding numerical solution of insect population model and Lotka-Volterra model. Here we apply two numerical methods called RKF and LADM for solutions of the said models. Here the numerical solutions obtained by using the RKF show high accuracy and these are compared with the LADM solution. So we can say that these numerical results show that the RKF method is an acceptable and reliable numerical technique for the solution of linear and nonlinear differential equation models on population models. It can be seen clearly from the graphical representations that RKF gives quite good results after a certain considerable time intervals. This is a very useful method, which will be undoubtedly found applicable in broad applications. The advantage of the RKF over the LADM is that there is no need for the evaluations of the Adomian polynomials and the advantage of RKF over RK4 is that it has a good accuracy using variable step size. Hence it provides an efficient numerical solution.

This dissertation focuses on developing efficient numerical methods and theoretical analysis for solving various inverse problems that arise in mathematics, physics, engineering, and beyond. We propose in this dissertation a unified framework with two stages to solve severely ill-posed and highly nonlinear inverse problems. In the first stage, we derive a system of partial differential equations by introducing a new variable and truncating the Fourier series of the solution to the governing equation. In the second stage, we solve the system derived in the first stage using the quasi-reversibility method, the Carleman contraction mapping method, and the convexification method. The obtained solutions of this stage directly yield the desired solutions to the inverse problems. An important contribution of the dissertation is that we will rigorously and numerically prove the efficiency of this framework, including its global convergence to the true solution. The analytic proofs are based on some Carleman estimates. The numerical proofs are provided by successfully testing our methods with highly noisy simulated data and experimental data provided by US Army Research Laboratory engineers.

We have no symbolic methods for solving this equation. Thus we will need to use the numerical methods mentioned above. One way we can get an idea of what the solutions to this (or any) differential equation look like is with a direction field. We've already seen in Assignment 2 how to use slopefield and drawode to get a picture of our solution.

Pictures like these are very informative for understanding the behavior of the solutions to even very complicated differential equations. Unfortunately, such graphical techniques are limited to three dimensions at best. Indeed, most technology problems involve high-dimensional systems of ODEs, so that it is impossible to visualize their solutions. Even with two- or three-dimensional systems, visual techniques might fail us when we want to get very precise information. For example, we might want to find out the value of y in our solution above when x = 10. From the direction field, we can guess that it is somewhere around 4.3. By changing the axes in slopefield, we might be able to zoom in and improve this estimate, but that can be an awkward way to obtain a highly accurate answer.

Euler's Method is one of the simplest and oldest numerical methods for approximating solutions to differential equations that cannot be solved with a nice formula. Euler's Method is also called the tangent line method, and in essence it is an algorithmic way of plotting an approximate solution to an initial value problem through the direction field. It can also be used to get an estimated value for that solution at a given value for x. Here is how it works: 041b061a72