\\n\\n
IntechOpen was founded by scientists, for scientists, in order to make book publishing accessible around the globe. Over the last two decades, this has driven Open Access (OA) book publishing whilst levelling the playing field for global academics. Through our innovative publishing model and the support of the research community, we have now published over 5,700 Open Access books and are visited online by over three million academics every month. These researchers are increasingly working in broad technology-based subjects, driving multidisciplinary academic endeavours into human health, environment, and technology.
\\n\\nBy listening to our community, and in order to serve these rapidly growing areas which lie at the core of IntechOpen's expertise, we are launching a portfolio of Open Science journals:
\\n\\nAll three journals will publish under an Open Access model and embrace Open Science policies to help support the changing needs of academics in these fast-moving research areas. There will be direct links to preprint servers and data repositories, allowing full reproducibility and rapid dissemination of published papers to help accelerate the pace of research. Each journal has renowned Editors in Chief who will work alongside a global Editorial Board, delivering robust single-blind peer review. Supported by our internal editorial teams, this will ensure our authors will receive a quick, user-friendly, and personalised publishing experience.
\\n\\n"By launching our journals portfolio we are introducing new, dedicated homes for interdisciplinary technology-focused researchers to publish their work, whilst embracing Open Science and creating a unique global home for academics to disseminate their work. We are taking a leap toward Open Science continuing and expanding our fundamental commitment to openly sharing scientific research across the world, making it available for the benefit of all." Dr. Sara Uhac, IntechOpen CEO
\\n\\n"Our aim is to promote and create better science for a better world by increasing access to information and the latest scientific developments to all scientists, innovators, entrepreneurs and students and give them the opportunity to learn, observe and contribute to knowledge creation. Open Science promotes a swifter path from research to innovation to produce new products and services." Alex Lazinica, IntechOpen founder
\\n\\nIn conclusion, Natalia Reinic Babic, Head of Journal Publishing and Open Science at IntechOpen adds:
\\n\\n“On behalf of the journal team I’d like to thank all our Editors in Chief, Editorial Boards, internal supporting teams, and our scientific community for their continuous support in making this portfolio a reality - we couldn’t have done it without you! With your support in place, we are confident these journals will become as impactful and successful as our book publishing program and bring us closer to a more open (science) future.”
\\n\\nWe invite you to visit the journals homepage and learn more about the journal’s Editorial Boards, scope and vision as all three journals are now open for submissions.
\\n\\nFeel free to share this news on social media and help us mark this memorable moment!
\\n\\n\\n"}]',published:!0,mainMedia:{caption:"",originalUrl:"/media/original/237"}},components:[{type:"htmlEditorComponent",content:'
After years of being acknowledged as the world's leading publisher of Open Access books, today, we are proud to announce we’ve successfully launched a portfolio of Open Science journals covering rapidly expanding areas of interdisciplinary research.
\n\n\n\nIntechOpen was founded by scientists, for scientists, in order to make book publishing accessible around the globe. Over the last two decades, this has driven Open Access (OA) book publishing whilst levelling the playing field for global academics. Through our innovative publishing model and the support of the research community, we have now published over 5,700 Open Access books and are visited online by over three million academics every month. These researchers are increasingly working in broad technology-based subjects, driving multidisciplinary academic endeavours into human health, environment, and technology.
\n\nBy listening to our community, and in order to serve these rapidly growing areas which lie at the core of IntechOpen's expertise, we are launching a portfolio of Open Science journals:
\n\nAll three journals will publish under an Open Access model and embrace Open Science policies to help support the changing needs of academics in these fast-moving research areas. There will be direct links to preprint servers and data repositories, allowing full reproducibility and rapid dissemination of published papers to help accelerate the pace of research. Each journal has renowned Editors in Chief who will work alongside a global Editorial Board, delivering robust single-blind peer review. Supported by our internal editorial teams, this will ensure our authors will receive a quick, user-friendly, and personalised publishing experience.
\n\n"By launching our journals portfolio we are introducing new, dedicated homes for interdisciplinary technology-focused researchers to publish their work, whilst embracing Open Science and creating a unique global home for academics to disseminate their work. We are taking a leap toward Open Science continuing and expanding our fundamental commitment to openly sharing scientific research across the world, making it available for the benefit of all." Dr. Sara Uhac, IntechOpen CEO
\n\n"Our aim is to promote and create better science for a better world by increasing access to information and the latest scientific developments to all scientists, innovators, entrepreneurs and students and give them the opportunity to learn, observe and contribute to knowledge creation. Open Science promotes a swifter path from research to innovation to produce new products and services." Alex Lazinica, IntechOpen founder
\n\nIn conclusion, Natalia Reinic Babic, Head of Journal Publishing and Open Science at IntechOpen adds:
\n\n“On behalf of the journal team I’d like to thank all our Editors in Chief, Editorial Boards, internal supporting teams, and our scientific community for their continuous support in making this portfolio a reality - we couldn’t have done it without you! With your support in place, we are confident these journals will become as impactful and successful as our book publishing program and bring us closer to a more open (science) future.”
\n\nWe invite you to visit the journals homepage and learn more about the journal’s Editorial Boards, scope and vision as all three journals are now open for submissions.
\n\nFeel free to share this news on social media and help us mark this memorable moment!
\n\n\n'}],latestNews:[{slug:"webinar-introduction-to-open-science-wednesday-18-may-1-pm-cest-20220518",title:"Webinar: Introduction to Open Science | Wednesday 18 May, 1 PM CEST"},{slug:"step-in-the-right-direction-intechopen-launches-a-portfolio-of-open-science-journals-20220414",title:"Step in the Right Direction: IntechOpen Launches a Portfolio of Open Science Journals"},{slug:"let-s-meet-at-london-book-fair-5-7-april-2022-olympia-london-20220321",title:"Let’s meet at London Book Fair, 5-7 April 2022, Olympia London"},{slug:"50-books-published-as-part-of-intechopen-and-knowledge-unlatched-ku-collaboration-20220316",title:"50 Books published as part of IntechOpen and Knowledge Unlatched (KU) Collaboration"},{slug:"intechopen-joins-the-united-nations-sustainable-development-goals-publishers-compact-20221702",title:"IntechOpen joins the United Nations Sustainable Development Goals Publishers Compact"},{slug:"intechopen-signs-exclusive-representation-agreement-with-lsr-libros-servicios-y-representaciones-s-a-de-c-v-20211123",title:"IntechOpen Signs Exclusive Representation Agreement with LSR Libros Servicios y Representaciones S.A. de C.V"},{slug:"intechopen-expands-partnership-with-research4life-20211110",title:"IntechOpen Expands Partnership with Research4Life"},{slug:"introducing-intechopen-book-series-a-new-publishing-format-for-oa-books-20210915",title:"Introducing IntechOpen Book Series - A New Publishing Format for OA Books"}]},book:{item:{type:"book",id:"325",leadTitle:null,fullTitle:"Alternative Fuel",title:"Alternative Fuel",subtitle:null,reviewType:"peer-reviewed",abstract:"Renewable energy sources such as biodiesel, bioethanol, biomethane, biomass from wastes or hydrogen are subject of great interest in the current energy scene. These fuels contribute to the reduction of prices and dependence on fossil fuels. In addition, energy sources such as these could partially replace the use of what is considered as the major factor responsible for global warming and the main source of local environmental pollution. For these reasons they are known as alternative fuels. There is an urgent need to find and optimise the use of alternative fuels to provide a net energy gain, to be economically competitive and to be producible in large quantities without compromising food resources.",isbn:null,printIsbn:"978-953-307-372-9",pdfIsbn:"978-953-51-4461-8",doi:"10.5772/851",price:139,priceEur:155,priceUsd:179,slug:"alternative-fuel",numberOfPages:360,isOpenForSubmission:!1,isInWos:1,isInBkci:!0,hash:"e6fd051beb23f469c73edd7bd22547bf",bookSignature:"Maximino Manzanera",publishedDate:"August 9th 2011",coverURL:"https://cdn.intechopen.com/books/images_new/325.jpg",numberOfDownloads:110121,numberOfWosCitations:133,numberOfCrossrefCitations:97,numberOfCrossrefCitationsByBook:5,numberOfDimensionsCitations:205,numberOfDimensionsCitationsByBook:6,hasAltmetrics:1,numberOfTotalCitations:435,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"November 8th 2010",dateEndSecondStepPublish:"December 6th 2010",dateEndThirdStepPublish:"April 12th 2011",dateEndFourthStepPublish:"May 12th 2011",dateEndFifthStepPublish:"July 11th 2011",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6,8",editedByType:"Edited by",kuFlag:!1,featuredMarkup:null,editors:[{id:"27371",title:"Dr.",name:"Maximino",middleName:null,surname:"Manzanera",slug:"maximino-manzanera",fullName:"Maximino Manzanera",profilePictureURL:"https://mts.intechopen.com/storage/users/27371/images/3570_n.jpg",biography:"Maximino Manzanera was born in Barcelona, Spain, in 1972. 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Maximino is participating actively in three main lines of research: Anhydrobiotic Engineering, Rhizoremediation and Biofuels of microbial origin.",institutionString:null,position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"University of Granada",institutionURL:null,country:{name:"Spain"}}}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"890",title:"Renewable Energy",slug:"environmental-sciences-waste-management-renewable-energy"}],chapters:[{id:"17583",title:"Biodiesel Production by Using Heterogeneous Catalysts",doi:"10.5772/23908",slug:"biodiesel-production-by-using-heterogeneous-catalysts",totalDownloads:16643,totalCrossrefCites:15,totalDimensionsCites:25,hasAltmetrics:1,abstract:null,signatures:"Rubi Romero, Sandra Luz Martínez and Reyna Natividad",downloadPdfUrl:"/chapter/pdf-download/17583",previewPdfUrl:"/chapter/pdf-preview/17583",authors:[{id:"54382",title:"Dr.",name:"Rubi",surname:"Romero",slug:"rubi-romero",fullName:"Rubi Romero"},{id:"57495",title:"MSc.",name:"Sandra Luz",surname:"Martínez",slug:"sandra-luz-martinez",fullName:"Sandra Luz Martínez"},{id:"57496",title:"Dr.",name:"Reyna",surname:"Natividad",slug:"reyna-natividad",fullName:"Reyna Natividad"}],corrections:null},{id:"17584",title:"An Overview of Enzyme-Catalyzed Reactions and Alternative Feedstock for Biodiesel Production",doi:"10.5772/24057",slug:"an-overview-of-enzyme-catalyzed-reactions-and-alternative-feedstock-for-biodiesel-production",totalDownloads:5652,totalCrossrefCites:1,totalDimensionsCites:8,hasAltmetrics:0,abstract:null,signatures:"Maria Manuela Camino Feltes, Débora de Oliveira, Jorge Luiz Ninow and José Vladimir de Oliveira",downloadPdfUrl:"/chapter/pdf-download/17584",previewPdfUrl:"/chapter/pdf-preview/17584",authors:[{id:"55230",title:"Dr.",name:"Maria Manuela",surname:"Camino Feltes",slug:"maria-manuela-camino-feltes",fullName:"Maria Manuela Camino Feltes"},{id:"57306",title:"Prof.",name:"Débora",surname:"De Oliveira",slug:"debora-de-oliveira",fullName:"Débora De Oliveira"},{id:"57307",title:"Prof.",name:"Jorge Luiz",surname:"Ninow",slug:"jorge-luiz-ninow",fullName:"Jorge Luiz Ninow"},{id:"57308",title:"Prof.",name:"Jose Vladimir",surname:"De Oliveira",slug:"jose-vladimir-de-oliveira",fullName:"Jose Vladimir De Oliveira"}],corrections:null},{id:"17585",title:"Biodiesel Fuel Production by Enzymatic Transesterification of Oils: Recent Trends, Challenges and Future Perspectives",doi:"10.5772/21905",slug:"biodiesel-fuel-production-by-enzymatic-transesterification-of-oils-recent-trends-challenges-and-futu",totalDownloads:11063,totalCrossrefCites:13,totalDimensionsCites:26,hasAltmetrics:0,abstract:null,signatures:"Nevena Luković, Zorica Knežević-Jugović and Dejan Bezbradica",downloadPdfUrl:"/chapter/pdf-download/17585",previewPdfUrl:"/chapter/pdf-preview/17585",authors:[{id:"45320",title:"MSc",name:"Nevena",surname:"Lukovic",slug:"nevena-lukovic",fullName:"Nevena Lukovic"},{id:"57079",title:"Prof.",name:"Dejan",surname:"Bezbradica",slug:"dejan-bezbradica",fullName:"Dejan Bezbradica"},{id:"57080",title:"Dr.",name:"Zorica",surname:"Knežević-Jugović",slug:"zorica-knezevic-jugovic",fullName:"Zorica Knežević-Jugović"}],corrections:null},{id:"17586",title:"Engine Test of Bio-Diesel Manufactured from Waste Cooking Oil and Reward Preferential Benefit Analyses for Its Promotion",doi:"10.5772/21712",slug:"engine-test-of-bio-diesel-manufactured-from-waste-cooking-oil-and-reward-preferential-benefit-analys",totalDownloads:2898,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,abstract:null,signatures:"Jai-Houng Leu",downloadPdfUrl:"/chapter/pdf-download/17586",previewPdfUrl:"/chapter/pdf-preview/17586",authors:[{id:"44408",title:"Dr.",name:"Jai-Houng",surname:"Leu",slug:"jai-houng-leu",fullName:"Jai-Houng Leu"}],corrections:null},{id:"17587",title:"Waste Animal Fats with High FFA as a Renewable Energy Source for Biodiesel Production - 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Numerical verification methods of solutions for differential equations have been the subject of extensive study in recent years and much progress have been made both mathematically and computationally [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. However, for some problems governed by the elliptic variational inequalities, there are very few approaches. As far as we know, it is hard to find any applicable methods except for those of Nakao and Ryoo [13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].
The authors have studied for several years the numerical verification method of solutions for elliptic variational inequalities using finite element method and the constructive error estimates combining with Schauder’s and Banach’s fixed point theorem. Several results in our research are already published in [13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In this chapter, we briefly overview our resent research results including works not yet published.
The outline of this chapter is as follows. In Section 2, the two types of elliptic variational inequalities are considered. In Subsection 2.1, we describe the elliptic variational inequalities and give a fixed point formulation to prove the existence of solutions. In Subsections 2.2 and 2.3, the main tool of the verification method is explained at an abstract level. In Subsection 2.2, we present a simple iteration method for numerical verification of solutions for the elliptic variational inequalities. We construct the concepts of rounding and rounding error for functions and present a computer algorithm to construct the set satisfying the verification conditions. However, it is difficult to apply the method in Subsection 2.2 to a problem in which an associated operator is not retractive in a neighborhood of the solution, because it is based upon a simple iteration method. In Subsection 2.3, we propose another approach to overcome such a difficulty. This method can be applied to general elliptic variational inequalities without any retraction property of the associated operator. We introduce a Newton-like operator and reformulate the problem using it. Particularly, special emphasis is placed on the way to devise the Newton-like operator for a kind of non-differentiable map which defines the original problem. We introduce a computational verification condition. In order to show a concrete usage of the tool, in Section 3, we present an application to some problems governed by the elliptic variational inequalities. Many difficulties remain to be overcome in the construction of general techniques applicable to a broader range of problems. However, the authors have no doubt that investigation along this line will lead to a new approach employing numerical methods in the field of existence theory of solutions for various variational inequalities that appear in mathematical analysis.
The theory of elliptic variational inequalities has become a rich source of inspiration in both mathematical and engineering sciences. Elliptic variational inequalities are an effective tool for studying the existence of solutions of constrained problems arising in mechanics, optimization and control, operation research, engineering science, etc. [47, 48, 49, 50, 51, 52]. It is the aim of this chapter to introduce a numerical technique to verify the solutions for elliptic variational inequalities. The basic approach of this technique consists of the fixed point formulation of elliptic variational inequalities and construction of the function set, on computer, satisfying the validation condition of a certain infinite dimensional fixed point theorem. For fixed point formulation, we consider a candidate set which possibly contains a solution. In order to get such a candidate set, we divide the verification procedure into two phases: one is the computation of a projection into a closed convex subset of some finite dimensional subspace (rounding); the other is the estimation of the error for the projection (rounding error). Combining these methods with some iterative technique, the exact solution can be enclosed by sum of rounding parts, which is a subset of finite dimensional space, and the rounding error, which is indicated by a nonnegative real number. These two procedures enable us to treat infinite dimensional problems as finite procedures, thta is, by computer.
A bilinear form
In general we do not assume
We consider two classes of elliptic variational inequalities.
Elliptic variational inequalities of the first kind:
Elliptic variational inequalities of the second kind:
Let us first set a few notations [1, 47, 49, 50, 53, 54, 55, 56, 57, 58, 59, 60, 61]. In what follows we shall make use of the Sobolev spaces
The norm in
Let
where
In order to obtain a fixed point formulation of variational inequality (1) we need the following standard result.
For some constant
where
For some constant
Using (4) in the above inequality, problem (1) is equivalent to that of finding
By (2) and (5), we now have the following fixed point problem for the operator
Under appropriate conditions on the space
In order to compute an explicit inclusion, we must therefore construct
If we restrict the nonlinear map
The problem (8) has the restricted condition; even so (8) is an important and very useful class of nonlinear problems arising in mathematical physics, mechanics, engineering sciences, etc. In Section 3, we briefly consider a particular example of interest in applications. Another example is given in [13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In the special case in which
where
In this subsection, we describe a simple iteration method for numerical verification of solutions for elliptic variational inequalities. In order to treat functions and variational inequalities in the infinite dimensional space
To verify the existence of a solution of (9), we determine a set
From Schauder’s fixed point theorem, if
Let us describe the procedure more concretely. First, we consider the auxiliary problem: given
We note that, by well known result [49], there is a unique element
Secondly, we define the approximate problem corresponding to (12) as
and (13) admit one and only one solution [49]. Error estimates for the variational inequalities can be found in [48, 49, 52], etc. Now, using (10), (12), (13) and error estimates, we make the following assumption.
In order to verify the solutions numerically, it is necessary to determine the constant
In order to construct the set
(1) First, we obtain an approximate solution
(2) Next we will define
holds for some
Here,
(3) Check the verification condition:
If the condition is satisfied, then
(4) If the condition is not satisfied, we continue the simple iteration by using
and then go back to the second step. The reader may refer to [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] for the details. If the condition (17) is satisfied, in our inclusion method of solutions for (9), the solution
The significance of a Newton-like operator was already pointed out in [29, 43]. Hence we will not discuss it in detail here. In Subsection 2.1, numerical verification of solutions for elliptic variational inequalities using a finite element method have been discussed only for simple iteration method. The method proposed in Subsection 2.2 is such that
In this subsection, we use the notation of Section 2.2. We assume that
Note that (9) can be rewritten as the following decomposed form in
In order to formulate a Newton-like verification condition for (18), we need a Fréchet derivative of the operator
To consider the Newton-like operator for (18), we define the nonlinear operator
Here
Next we define the operator
Then
When we decompose the set
Here,
respectively.
Note that
Therefore, the former condition in (21) is validated as the inclusion relations of corresponding coefficient intervals; the latter part can be checked by comparing two nonnegative real numbers.
Next we show a computer algorithm to construct the set
Finally, the verification condition in a computer is given by the following theorem. The proof of Theorem 4 will be given here for the sake of completeness; it is based on Proposition 3 and Schauder’s fixed point theorem.
hold, then there exists a solution
The study for the numerical verification method for elliptic variational inequalities has been still made less progress than for the differential equation case. The author’s method in the present chapter can be also applied, in principal, to the verification of solutions of the practical problems. Namely, in Section 3.1, we first give, a slightly detailed descroption of the basic principle and formulation of our numerical verification method for the solution of obstacle problems with a homogeneous condition. This should be an appropriate introduction to another applications of our idea. The basic approach of the method consisits of the fixed point formulation of the problems and construction of the function set, in a computer, satisying the validation condition of a certain infinite dimensional fixed point theorem. We also mention that it is possible to extend the method to more general problems with non-homogenerous obstacles. Moreover, in order to apply our method to the problem whose associated operator is not retractive in a neighborhood of the solution, a Newton-like method is introduced. Next, in Section 3.2, we apply our method to another type of free boundary problem with appears in the elasto-plastic deformation theory. This problem causes some properties of non-smoothness in tha associated finite dimensional equations. But, we can also overcome such a difficulty by appling the solution method for non-smooth problems developed by [29, 32, 33]. In the Section 3.3, we briefly remark that our enclosure method can also be applied to the so-called simplified Signorini problem which is a simplified version of a problem accurring in the elasticity theory [43]. Finally, in Section 3.4, we show the way to apply our approach to elliptic variational inequalities of the second kind appearing in the flow problems of a viscos-plastic fulied in a pipe.
We introduce the verification method for solutions of the obstacle problem which is known as a free boundary problem to cahracterize the contacted zone by an obstacle
Here, ‘homogeneous’ stands for the case that obstacle
Though the basic idea of verification is given in other places [26, 27, 28], in order to keep the paper as self-contained as possible, we describe rather detailed formulation and verification procedure for the present case.
Let
which is adopted as the inner product on
First, we note that, by well-known result [49], for any
has a unique solution
holds [49], where
Now consider the following elliptic variational inequalities with nonlinear right-hand side;
We take an appropriate finite dimensional subspace
We also define the projection
And define the projection
Now, as one of the approximation properties of
For each
Here,
First, note that, for each
which also implies that
Then the map
Using the definition (27) and (32), we now have the following fixed point problem for the compact operator
We introduce two concepts, rounding and rounding error, which enable us to deal with the infinite dimensional problem by finite procedures, that is, in a computer.
Now we define the dual cone of
and note that
For any
Next, for any subset
Usually,
Moreover, for
where
Here,
Thus we may take as
Therefore, the following verification condition is obtained by Schauder’s fixed point theorem.
then there exists a solution of
We sometimes refer the above set
We describe the method to find a set
Consider the following approximate solution
Since the bilinear form
Let
where
where
Furthermore, we define for any
Then, for a given candidate set
Here,
Thus we can present a computational verification condition. In the actual computation, we use an iterative procedure with
In this subsection, we consider the two-dimensional case. In order to verify solutions numerically, it is necessary to determine some constants that appear in the a priori error estimates. For the non-homogeneous case, we define
Note that, in general,
where,
We provide a numerical example of verification in the two-dimensional case according to the procedures described in the previous section. Let
The execution conditions are as follows (Figures 1–3):
Obstacle function
Approximate solution
Illustration of the contact zone.
Results are as follows:
Detailed arguments and with numerical examples are presented in [42].
The idea of the enclosure method for solutions of obstacle problems is based upon simply sequential iterations for the original fixed point operator
To formulate a Newton-type verification condition, we need a Fréchet derivative of the operator
where
And, for
Here,
And let
In order to define
Here,
Then we define the approximate Fréchet-like derivative of
We now assume that.
Here,
By using the above approximate Fréchet-like derivative, we define the Newton-like operator
Next we define the operator
Then
In this subsection, we consider an enclosure metnod of solutions for elasto-plastic torsion problems governed by an elliptic variational inequalities [25, 32, 33]. The nonlinear elasto-plastic torsion problem is defined as the same type elliptic variational inequalities as (26) with
As is well known [56, 58], two sub-domains
and
correspond to the plastic and elastic regions, respectively. The elastic region
The finite dimensional convex subset
In order to formulate the verification procedure, we need a verified computational method for solving the finite dimensional part (rounding) and a constructive estimates for infinite dimensional part (rounding error) as in the previous subsection.
Following [49, 56], we define the Lagrangian functional
It follows, from [49, 56], that if
The problem (46) can be formulated as a system of nonlinear and nonsmooth (nondifferentiable) equations. A verification method for nonsmooth equations by a generalized Krawczyk operator is studied in [1, 55]. We briefly describe the method presented by [55] in the below.
We consider the following equivalent system of nonlinear(and nondifferentiable) equation to (46) for a fixed
Here, we assume that
where [x] stands for an interval vector, “co” denotes the convex hull, and
Let
Then an interval operator for nonsmooth equations is defined by
The mapping
Thus, we can compute the solution of (46) with guaranteed accuracy. That is, we can enclose the rounding
A simplified Signorini problem is also given by the elliptic variational inequalities of the form (26) with
and
where
As well known, the solution
where
For a candidate set
Since the constant
In this subsection, we show that our idea of verification method can also be applied to the elliptic variational inequalities of the second kind.
Now, we define the functional
As in the previous section, we consider the following auxiliary problem associated with (53) for a given
By the well known result, we have the following lemma.
When we denote the solution
With the above inequalities, we obtain
Therefore, we obtain
That is,
We now define the approximate problem corresponding to (54) as
In order to apply our verification method to enclose the solutions of (53), we need a guaranteed computation of the exact solution of the problem (55),
A major difficulty in solving the problem (55) numerically is the processing of the nondifferentiable term
Let us define
Moreover, it is known that (56) is equivalent to the following problem:
Here
where
Then our first purpose, computing the rounding
The Eq. (58) leads to a kind of finite dimensional, nonlinear but nondifferentiable problem. We use a slope function method proposed by Rump [18, 19, 20] to enclose the solutions of (58) with
Here, we may take
We have surveyed numerical verification methods for differential equations, especially around partial differential equations, variational inequalities and the author’s works. But the period of this research is shorter than the history of the numerical methods for differential equations by computer and we can say it is still in the stage of case studies. Indeed, recently, this kind of studies have been referred little by little for practical applications in PDEs and variational inequalities but there are many open problems to be resolve. Therefore, we can make no safe prediction that these approaches will grow into really useful methods for various kinds of equations and variational inequalities in mathematical analysis. Also, since the program description of the verification algorithm is very complicated in general, there is another problem like software technology associated with assurance for the correctness of the verification program itself. Actually, some of the mathematician would not give credit the computer assisted proof in analysis as correct as they believe the theoretical proof, which might cause a kind of seriously emotional problem in the methodology of mathematical sciences. And there is another difficulty from the huge scale of numerical computations which often exceed the capacity of the concurrent computing facilities.
However, in the twenty-first century, the computing environment would make more and more rapid progress, which should be beyond conception in the present state. In any case, a realistic study for partial differential equations and variational inequalities should be the future subject of the numerical computations with guaranteed accuracy. The authors believe that numerical methods with guaranteed accuracy for differential equations and variational inequalities would highly improve the reliability in the numerical simulation of the complicated phenomena in both mathematical and engineering sciences.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1A2B4006092).
65N15; 65G20
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\\n\\nAlthough such papers are regularly made publicly available via personal websites and institutional repositories, their general purpose is to gather comments and feedback from Authors’ colleagues in order to further improve a manuscript intended for future publication.
\\n\\nWhen submitting their work, Authors are required to disclose the existence of any publicly available earlier drafts in a note to the Academic Editor. In cases where earlier drafts of the submitted version of the manuscript are publicly available, any overlap between the versions will generally not be considered an instance of self-plagiarism.
\\n\\n4. SOCIAL MEDIA, BLOG & MESSAGE BOARD POSTINGS
\\n\\nWe feel that social media, blogs and message boards are generally used with the same intention as grey literature, to formulate ideas for a manuscript and gather early feedback from like-minded researchers in order to improve a particular piece of work before submitting it for publication. Therefore, we do not consider such internet postings to be publication in the scholarly sense.
\\n\\nNevertheless, Authors are encouraged to disclose the existence of any internet postings in which they outline and describe their research or posted passages of their manuscripts in a note to the Academic Editor. Please note that we will not strictly enforce this request in the same way that we would instructions we consider to be part of our conditions of acceptance for publication. We understand that it may be difficult to keep track of all one’s internet postings in which the researcher´s current work might be mentioned.
\\n\\nIn cases where there is any overlap between the Author´s submitted manuscript and related internet postings, we will generally not consider it to be an instance of self-plagiarism. This also holds true for any co-Author as well.
\\n\\nFor more information on this policy please contact permissions@intechopen.com.
\\n\\nPolicy last updated: 2017-03-20
\\n"}]'},components:[{type:"htmlEditorComponent",content:'A significant number of working papers, early drafts, and similar work in progress are openly shared online between members of the scientific community. It has become common to announce one’s own research on a personal website or a blog to gather comments and suggestions from other researchers. Such works and online postings are, indeed, published in the sense that they are made publicly available. However, this does not mean that if submitted for publication by IntechOpen they are not original works. We differentiate between reviewed and non-reviewed works when determining whether a work is original and has been published in a scholarly sense or not.
\n\nThe significance of Peer Review cannot be overstated when it comes to defining, in our terms, what constitutes a published scientific work. Peer Review is widely considered to be the cornerstone of modern publishing processes and the key value-adding contribution to a scholarly manuscript that a publisher can make.
\n\nOther than the issue of originality, research misconduct is another major issue that all publishers have to address. IntechOpen’s Retraction & Correction Policy and various publication ethics guidelines identify both redundant publication and (self)plagiarism to fall within the definition of research misconduct, thus constituting grounds for rejection or the issue of a Retraction if the work has already been published.
\n\nIn order to facilitate the tracking of a manuscript’s publishing history and its development from its earliest draft to the manuscript submitted, we encourage Authors to disclose any instances of a manuscript’s prior publication, whether it be through a conference presentation, a newspaper article, a working paper publicly available in a repository or a blog post.
\n\nA note to the Academic Editor containing detailed information about a submitted manuscript’s previous public availability is the preferred means of reporting prior publication. This helps us determine if there are any earlier versions of a manuscript that should be disclosed to our readers or if any of those earlier versions should be cited and listed in a manuscript’s references.
\n\nSome basic information about the editorial treatment of different varieties of prior publication is laid out below:
\n\n1. CONFERENCE PAPERS & PRESENTATIONS
\n\nGiven that conference papers and presentations generally pass through some sort of peer or editorial review, we consider them to be published in the accepted scholarly sense, particularly if they are published as a part of conference proceedings.
\n\nAll submitted manuscripts originating from a previously published conference paper must contain at least 50% of new original content to be accepted for review and considered for publication.
\n\nAuthors are required to report any links their manuscript might have with their earlier conference papers and presentations in a note to the Academic Editor, as well as in the manuscript itself. Additionally, Authors should obtain any necessary permissions from the publisher of their conference paper if copyright transfer occurred during the publishing process. Failure to do so may prevent Us from publishing an otherwise worthy work.
\n\n2. NEWSPAPER & MAGAZINE ARTICLES
\n\nNewspaper and magazine articles usually do not pass through any extensive peer or editorial review and we do not consider them to be published in the scholarly sense. Articles appearing in newspapers and magazines rarely possess the depth and structure characteristic of scholarly articles.
\n\nSubmitted manuscripts stemming from a previous newspaper or magazine article will be accepted for review and considered for publication. However, Authors are strongly advised to report any such publication in an accompanying note to the External Editor.
\n\nAs with the conference papers and presentations, Authors should obtain any necessary permissions from the newspaper or magazine that published the work, and indicate that they have done so in a note to the External Editor.
\n\n3. GREY LITERATURE
\n\nWhite papers, working papers, technical reports and all other forms of papers which fall within the scope of the ‘Luxembourg definition’ of grey literature do not pass through any extensive peer or editorial review and we do not consider them to be published in the scholarly sense.
\n\nAlthough such papers are regularly made publicly available via personal websites and institutional repositories, their general purpose is to gather comments and feedback from Authors’ colleagues in order to further improve a manuscript intended for future publication.
\n\nWhen submitting their work, Authors are required to disclose the existence of any publicly available earlier drafts in a note to the Academic Editor. In cases where earlier drafts of the submitted version of the manuscript are publicly available, any overlap between the versions will generally not be considered an instance of self-plagiarism.
\n\n4. SOCIAL MEDIA, BLOG & MESSAGE BOARD POSTINGS
\n\nWe feel that social media, blogs and message boards are generally used with the same intention as grey literature, to formulate ideas for a manuscript and gather early feedback from like-minded researchers in order to improve a particular piece of work before submitting it for publication. Therefore, we do not consider such internet postings to be publication in the scholarly sense.
\n\nNevertheless, Authors are encouraged to disclose the existence of any internet postings in which they outline and describe their research or posted passages of their manuscripts in a note to the Academic Editor. Please note that we will not strictly enforce this request in the same way that we would instructions we consider to be part of our conditions of acceptance for publication. We understand that it may be difficult to keep track of all one’s internet postings in which the researcher´s current work might be mentioned.
\n\nIn cases where there is any overlap between the Author´s submitted manuscript and related internet postings, we will generally not consider it to be an instance of self-plagiarism. This also holds true for any co-Author as well.
\n\nFor more information on this policy please contact permissions@intechopen.com.
\n\nPolicy last updated: 2017-03-20
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