Pub Date : 2019-04-05DOI: 10.1002/9781119423461.ch8
{"title":"Boundary Element Method","authors":"","doi":"10.1002/9781119423461.ch8","DOIUrl":"https://doi.org/10.1002/9781119423461.ch8","url":null,"abstract":"","PeriodicalId":366025,"journal":{"name":"Advanced Numerical and Semi-Analytical Methods for Differential Equations","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117325999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-05DOI: 10.1002/9781119423461.ch11
S. Chakraverty, N. Mahato, P. Karunakar, T. D. Rao
The Adomian decomposition method (ADM) is an efficient semi‐analytical technique used for solving linear and nonlinear differential equations. It permits us to handle both nonlinear initial value problems (IVPs) and boundary value problems. The solution technique of this method is mainly based on decomposing the solution of nonlinear operator equation to a series of functions. Each presented term of the obtained series is developed from a polynomial generated in the expansion of an analytic function into a power series. This chapter presents procedures for solving linear as well as nonlinear ordinary/partial differential equations by the ADM along with example problems for clear understanding. It also presents linear and nonlinear IVPs for clear understanding of the ADM for ordinary differential equations. ADM transforms system of partial differential equations into a set of recursive relation that can easily be handled. To understand the method, one can consider the system of linear partial differential equations.
{"title":"Adomian Decomposition Method","authors":"S. Chakraverty, N. Mahato, P. Karunakar, T. D. Rao","doi":"10.1002/9781119423461.ch11","DOIUrl":"https://doi.org/10.1002/9781119423461.ch11","url":null,"abstract":"The Adomian decomposition method (ADM) is an efficient semi‐analytical technique used for solving linear and nonlinear differential equations. It permits us to handle both nonlinear initial value problems (IVPs) and boundary value problems. The solution technique of this method is mainly based on decomposing the solution of nonlinear operator equation to a series of functions. Each presented term of the obtained series is developed from a polynomial generated in the expansion of an analytic function into a power series. This chapter presents procedures for solving linear as well as nonlinear ordinary/partial differential equations by the ADM along with example problems for clear understanding. It also presents linear and nonlinear IVPs for clear understanding of the ADM for ordinary differential equations. ADM transforms system of partial differential equations into a set of recursive relation that can easily be handled. To understand the method, one can consider the system of linear partial differential equations.","PeriodicalId":366025,"journal":{"name":"Advanced Numerical and Semi-Analytical Methods for Differential Equations","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115256627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-05DOI: 10.1002/9781119423461.ch19
{"title":"Differential Equations with Interval Uncertainty","authors":"","doi":"10.1002/9781119423461.ch19","DOIUrl":"https://doi.org/10.1002/9781119423461.ch19","url":null,"abstract":"","PeriodicalId":366025,"journal":{"name":"Advanced Numerical and Semi-Analytical Methods for Differential Equations","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127615453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-06-09DOI: 10.1002/9781119423461.ch13
B. Batiha
Differential equations are encountered in various fields such as physics, chemistry, biology, mathematics and engineering. Most nonlinear models of real-life problems are still very difficult to solve either numerically or theoretically. Many unrealistic assumptions have to be made to make nonlinear models solvable. There has recently been much attention devoted to the search for better and more efficient solution methods for determining a solution, approximate or exact, analytical or numerical, to nonlinear models. Finding exact/approximate solutions of these nonlinear equations are interesting and important. One of these methods is variational iteration method (VIM), which has been proposed by Ji-Huan He in 1997 based on the general Lagrange’s multiplier method. The main feature of the method is that the solution of the linearized problem is used as the initial approximation for the linear and nonlinear problems. Then a more highly precise approximation at some special point can be obtained. This approximation converges rapidly to an accurate solution. VIM is very powerful and efficient in finding analytical as well as numerical solutions for a wide class of differential equation
{"title":"Variational Iteration Method","authors":"B. Batiha","doi":"10.1002/9781119423461.ch13","DOIUrl":"https://doi.org/10.1002/9781119423461.ch13","url":null,"abstract":"Differential equations are encountered in various fields such as physics, chemistry, biology, mathematics and engineering. Most nonlinear models of real-life problems are still very difficult to solve either numerically or theoretically. Many unrealistic assumptions have to be made to make nonlinear models solvable. There has recently been much attention devoted to the search for better and more efficient solution methods for determining a solution, approximate or exact, analytical or numerical, to nonlinear models. Finding exact/approximate solutions of these nonlinear equations are interesting and important. One of these methods is variational iteration method (VIM), which has been proposed by Ji-Huan He in 1997 based on the general Lagrange’s multiplier method. The main feature of the method is that the solution of the linearized problem is used as the initial approximation for the linear and nonlinear problems. Then a more highly precise approximation at some special point can be obtained. This approximation converges rapidly to an accurate solution. VIM is very powerful and efficient in finding analytical as well as numerical solutions for a wide class of differential equation","PeriodicalId":366025,"journal":{"name":"Advanced Numerical and Semi-Analytical Methods for Differential Equations","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124011836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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