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Carroll R. This specific ISBN edition is currently not available. View all copies of this ISBN edition:. Synopsis Shipped from UK, please allow 10 to 21 business days for arrival. Learn more about this copy. Other Popular Editions of the Same Title. Search for all books with this author and title. Ships in a sturdy cardboard container with tracking!. Published by Dover Publications Inc. About this Item: Dover Publications Inc. Language: English. Brand new Book. Detailed and self-contained, this treatment is directed to graduate students with some previous exposure to classical partial differential equations.
The author examines a variety of modern abstract methods in partial differential equations, especially in the area of abstract evolution equations.
Additional topics include the theory of nonlinear monotone operators applied to elliptic and variational problems. Book is in Used-Good condition. Since applications problems in science and engineering often lead to higher-index PDAEs, new techniques are required to solve these problems efficiently. Nevertheless, the power series method PSM Forsyth ; Ince is a well-known classic straightforward procedure from literature that can be applied successfully to solve differential equations of different kind: linear ordinary differential equations ODEs Coddington ; Forsyth ; Ince ; Kreyszig , nonlinear ODEs Biazar et al.
This method establishes that the solution of a differential equation can be expressed as a power series of the independent variable. Finally, we take the inverse LT of the PA to obtain the analytical solution. The proposed method does not produce noise terms also known as secular terms as the homotopy perturbation based techniques Soltanian et al. This greatly reduces the volume of computation and improves the efficiency of the method in comparison to the perturbation based methods.
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The rest of this paper is organized as follows. In the next section we illustrate the basic concept of the PSM.go site
ISBN 13: 9780060411978
Finally, a conclusion is drawn in the last section. Equation 3 is substituted into 1 , then we regroup the equation in terms of powers of t. The boundary conditions of 1 are substituted into 3 to generate an algebraic equation for each boundary condition. Finally, we solve the algebraic linear system to obtain the coefficients u 0 , u 1 ,…. Several approximate methods provide power series solutions polynomial. Nevertheless, sometimes, this type of solutions lacks of large domains of convergence.
L and M are arbitrarily chosen, but they should be of smaller value than the order of the power series. Finally, by using the inverse Laplace s transformation, we obtain the exact or approximate solution.
Abstract Methods in Partial Differential Equations
We assume that the solution to initial boundary-value problem 10 - 13 exists, is unique and sufficiently smooth. It is important to note that the time integration of equation 10 is not relevant to the solution procedure presented here, so one can apply the PSM directly to Then substitute 15 into system 11 - 14 and equate the coefficients of powers of t in the resulting polynomial equations to zero to get an algebraic linear system for these coefficients. The solutions series obtained from PSM may have limited regions of convergence, even if we take a large number of terms.
In this section, we will demonstrate the effectiveness and accuracy of the LPPSM presented in the previous section through two PDAE systems of index-one and index-three. Since one time differentiation of equation 17 determines u 2 t in terms of u and its space derivatives, then PDAE 16 - 17 is index-one.
Numerical Methods for Partial Differential Equations: Early View
Note that no initial condition is prescribed for the variable u 2 as this is determined by the PDAE. The solutions series obtained from the PSM may have limited regions of convergence, even if we take a large number of terms.
First, we apply t -Laplace transform to 28 and Since three time differentiations of equation 40 determine u 3 t in terms of the solution u and its space derivatives, then PDAE 38 - 40 is index-three. Therefore, this PDAE is difficult to solve numerically. Moreover no initial condition is prescribed for the variable u 3 as this is determined by the PDAE.
First t -Laplace transform is applied to 54 , 55 and In this paper we presented the power series method PSM as a useful analytical tool to solve partial differential-algebraic equations PDAEs. Two PDAE problems of index-one and index-three were solved by this method leading to the exact solutions. The method has successfully handled the index-three PDAE without the need for a preprocessing step of index-reduction.
For each of the two problems solved here, the PSM transformed the PDAE into an easily solvable linear algebraic system for the coefficient functions of the power series solution. These cookies allow us to monitor OverDrive's performance and reliability. They alert us when OverDrive services are not working as expected. Without these cookies, we won't know if you have any performance-related issues that we may be able to address.
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