الفهرس 
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Abstract
Interested in the subject of numerical analysis and the theory of approximation to provide approximate methods for solution of issues formulated mathematically, such as differential equations, and, since there are many natural phenomena, which can be described using these equations or Bonamatha integration debate, we will take care of in this letter and in particular classes of differential equations linear and nonlinear , and we will focus here on access to Altaqarib explicit knowledge of the functions of such equations, the differential equations of third and fifth grades. As we focus on the differential equations of any rank linear or non-linear terms of marginal homogeneous and heterogeneous.
Is the most important goals in this letter, which consists of four chapters, is to provide and develop new algorithms and efficient for direct solutions of differential equations of rank third, fifth and synonyms integration, using the method of Petrov - Jalrkin on the basis of the spectral expression for the solution spectrum in terms of many of the limits of Jacobi. These methods rely mainly on the establishment of rules independent of functions linear combination of many of the limits of Jacobi, then loosen functions, then the solution in terms of these rules, which we can apply the method of Petrov - Jalrkin Aalmadelat the differential or complementary patterns of differential equations to be solved. This results in construction to the appointment of transactions Mphakik solutions through systems, written with matrices of building a special, which we can find a Mekosadtha effectively and efficiently.
Also aim in this letter to make and build algorithms effective depends on Taqarib way Jintao for solutions Altgaribh equations differential with the initial conditions of constant high-level transactions. We aim also to create new algorithms and efficient depends on how the aggregate for solutions Altgaribh of differential equations with initial conditions of the higher levels of non-linear and that terms of many of the limits of Gacopy shifted, this causes construction to set the transactions Mphakik solutions through systems algebraic equations Gertih.
Mufkokat we proposed solutions to obtain the required approximations for any possible value of the pulling
(α> -1, β> -1). We examine, in particular cases, three special and important and is using many of the limits of Tshipihv type I α = β = -1 / 2)) and second (α = β = 1 / 2), and many limits to Ajindar (α = β = 0) . As well as two special cases within Pkthirat Chbaishev type III and IV (2 / 1 ± = β-= α). The results clarified the theory and numerical as well as that systems based on the loose in terms of polynomial Chbaishev type I α = β = -1 / 2)) is not the best from the rest of the limits of many other Jacobi.
In the first chapter we gave a brief introduction to roads spectral characteristics and the finite difference methods and ways of ending element. Also made clear the differences between the three spectral methods and is commonly used roads Jalrkin and the way in the aggregate and the way Jintao. We also give a brief study of orthogonal polynomials and their properties and functions Mphakik Bdalaltha. We have given as well as some general characteristics of many of the limits of Jacobi.
We have in the second quarter and in detail Khorazmyat new and effective to solve equations of the third and fifth grade fixed transactions in the cases of whether the boundary conditions homogeneous and heterogeneous using Jacobi Petrov - Jalrkin. Also used the method of Jacobi - Jacobi Petrov - Jalrkin to solve equations of rank third and fifth transactions variable.
We discussed in Chapter III how to apply the algorithms proposed in the second quarter to resolve patterns of integration corresponding to the differential equations of rank third and fifth by agents fixed and the same conditions of marginal homogeneous and heterogeneous . We discussed also how to mix my way Petrov - Jalrkin and aggregate to resolve patterns of integration corresponding to the differential equations of rank third and fifth transactions variable and the same conditions of marginal homogeneous and heterogeneous.
Finally, in the fourth quarter we have a way Jintao in terms of many of the limits of Jacobi shifted to solve equations Altfadilh with the initial conditions of higher levels of sin, and we have proposed as well as how many aggregate limits in terms of shifted Jacobi equations to solve Altfadilh senior-level non-linear.
we explain the results obtained in this letter in the form of charts and pictograms whenever we can. These results clarified that the proposed algorithms to find approximate solutions of the spectral differential equations and the corresponding patterns of integration that we have studied a minute.
and to our knowledge, the formulas and algorithms that were given in Chapters II and even the fourth entirely new.
which requires male and the programs that were used in this letter carried on the personal computer of the type (Intel (R) Core (TM) 2 Duo CPU 2.00 GHz, 2.00 GB of RAM) as well as we have with the software, known as the symbolic arithmetic (Mathematica 5.1) for the work of intermediate calculations and spreadsheets, as well as the illustrations in the message as a whole.