Evaluating Volterra Integro-Differential Equations in terms of Global, Polynomial and Numerical Equations in Boundary Conditions
The integro-differential equations are originated from different mathematical models for many scientific phenomena. Nonlinear integro-differential equations are also can be seen in various applications of various scientific fields that are modeled by nonlinear phenomena.[3]
The solutions using in integro-differential equations have an important role in lots of engineering fields, also in financial problems, physics theories. The major area of integro-differential –equations are especially mechanical engineering, electric-electronic engineering, economics.[5] Boundary conditions are very important for Volterra equations in order to make them more visual. Furthermore the benefit working on boundary conditions is to see excellent satability properties and high accuracy for Volterra equations.[1] In addition, while evaluating integro differential equations, we should consider the situations about nonlinear integro-differential equations. Nonlinear integro differential equations are essential also in several fields. For instance, fluid dynamics, polymer science, population dynamics, thermoelasticity, chemical engineering can be researching area.[2]
Because to solve a problem analytically can be very hard and spend a lot of time, global, polynomial and numerical methods can be very useful. However, in last decades, numerical methods have been used by many scientists. These numerical methods can be listed like The Taylor-series expansion method, the hybrid function method, Adomian decomposition method, The Legendre wavelets method, The Tau method, The finite difference method, The Haar function method, The...
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...initial value problems. To acquire global solution for differential equations in general, the concept of fuzzy linear differential equation is utilized. [6]
To conclude, with the increasing development which is used in the engineering, mechanics, physics, chemistry and astronomy, integro-differential equations are an indispensable major of mathematics. One of the most important equations, Volterra integro differential equations have various solutions by global, polynomial and numerical. These solutions have different various advantages and disadvantages for using different scientific fields. While one type of a solution is useful engineering, another solution is better for astronomy. Because of these reasons, to know every detail about methods for Volterra integro equations, to comprehend the principles of equations for being a good scientist is very essential.
i.e. K ̇(t)=sY(t)-δK(t), L ̇(t)=nL(t) and A ̇(t)=gA(t) it is important to consider the new assumptions that concern the newly added inputs.
The weakest feature of the paper is that although the formulas, presented by authors, are in general correct, but they do not support the conclusions the author extract from them, and mistake is hidden in the interpretation.
One of the researchers was Shu that he presented the differential quadrature method (DQ) for the first time [71]. To solve the computational mechanic problems, Bert and Malik [72] applied the DQM and they reported that the DQM was the most efficient method and they reported that the DQM solved the differential equations with fewer grid points and high accuracy. To this end, to date, the researchers have used the DQM to solve many mechanical engineering problems [73-80]. It is caused to gain accurate results in the lowest time. As noted, supplying the results which have the lowest grid point and deriving in the fast way are two important advantages of DQ method. In the recent years, to obtain the mechanical properties of the nanostructures the DQM is used and this is caused to produce the mechanical property of these structures accurately. The stability behavior of the rectangular nanoplate is investigated by Mohammadi et al. [77]. In that research, the DQM is used to obtain the shear in-plane buckling loads of the rectangular nanoplate. The stability analysis of the rectangular nanoplate subjected to linear load is investigated by Farajpour et al.
The Arrhenius equation ln k = ln A – (Ea / RT) can be shown
method can be produced and a graph of the function can be made. From the graph,
The dynamic systems view was developed by Arnold Gesell in 1934 and explores how humans develop their motor skills. From Mr. Gesell’s observations, he was able to conclude that children develop their motor skills in a specific order and time frame. He concluded that children roll, walk, sit, and stand as a result of several factors – the ability to move, the environmental support to move and the motivation/goal to move. Once the child has the motivation, ability, and support, they accept the new challenge. After several failed and successful attempts, they begin to fine-tune and master the movement with continued support and motivation. The dynamic systems theory is not a random process that children experience, the skills are calculated and develop over a period of time.
Trask, J. Chapter 8 Alternative Methods [Power Point slides]. Retrieved from Lecture Notes Online Website: https://compass.illinois.edu/webct/urw/lc5116011.tp0/cobaltMainFrame.dowebct
The calculus topic I would like to discuss comes from unit two, derivatives. Derivatives are enjoyable because in most cases, they are simple to solve. Also, derivatives make other classes involving calculus and derivatives easier to understand. Within this paper, I will be elaborating on differentiation, the derivative, rate of change, the rules and purpose of derivatives and how to understand them.
Second, the terza rima scheme helps the narrator to express his thoughts. In A Defense of Poetry, Shelley states that there exists harmony between the language that poets employ and the sounds that are contained in each word because both sounds and thoughts are intertwined to convey the message that they attempt to represent (763). In other words, there exists a close proximity between the sense of words and their sound; it is the enchainment of both ideas and sounds that creates an effect of harmony. Thus, language and sound work in conjunction to create a stylized and harmonious message which comes to life each time the poem is either read or recited out loud. We mentioned earlier that the Ode to the West Wind is an ode that addresses, that
On a more scientific note I am interested in mechanics of fluids. This interest was enforced last year when I had the opportunity to attend a lecture on fluid mechanics at P&G. At the conference I greatly expanded my knowledge regarding the physical aspect of fluids and their properties. In last year's AS course we have met a topic in this field. I will be applying ideas and knowledge gathered from last year for this investigation.
"Properties of sine waves." University of Manitoba. University of Manitoba, 2010. Web. 29 Nov. 2013. .
Differential analysis is useful in many situations faced by the management and it has to choice between different alternatives for each situation to make the necessary decision. Some of these situations are as follows:
Chapter one presents the Introduction. It also contains the statement of the problem and its signi...
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
Asymptotic analysis is a key tool to study nonlinear difference equations which arise in the mathematical modelling of real-world phenomena. It is not expected that explicit solutions can be found for the solutions of nonlinear difference equations; however, some nonlinear equations can be transformed into equivalent linear equations by a change of dependent variable. In this work, we transform a discrete logistic equation, which is a nonlinear difference equation, into a linear equation and we determine its explicit solution. This result able us to study the behavior of this solution and check the results known in stability theory.