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.
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 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.
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 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.
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...
... model for the thermodynamics and fluid mechanics calculations for this system need to be presented.
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
This equation is the simplest nonlinear first-order difference equation. In spite of its simplicity, this equation exhibits complicated dynamics. If we know the initial population given by then we find its solution, by simple iteration we it is the orbit of
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.
Currently present are non invasive computational modelling techniques, these are very informative and display results with high levels of accuracy but they do have some drawbacks. It requires a long and dreary approach of formulating math equations which must then be followed by substantial computational efforts. Besides that, model preparations also involve time consuming efforts. In order to minimize computational run-time, certain assumptions have to be made in order to simplify the governing equations and hence expose a certain uncertainty in the results generated.
where N_k are the constants and can be determined by the initial values of q_j and q ̇_j.
"Properties of sine waves." University of Manitoba. University of Manitoba, 2010. Web. 29 Nov. 2013. .