Problem Solving Basics for One Dimensional Kinematics

Problem Solving Basics for One Dimensional Kinematics tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập l...

RESEARC H Open Access An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab Natalia Nieves Salva 1,2 , Domingo Alberto Tarzia 1,3* and Luis Tadeo Villa 1,4 * Correspondence: DTarzia@austral. edu.ar 1 CONICET, Rosario, Argentina Full list of author information is available at the end of the article Abstract Nonlinear problems for the one-dimensional heat equation in a bounded and homogeneous medium with temperature data on the boundaries x = 0 and x = 1, and a uniform spatial heat source depending on the heat flux (or the temperature) on the boundary x = 0 are studied. Existence and uniqueness for the solution to non-classical heat conduction problems, under suitable assumptions on the data, are obtained. Comparisons results and asymptotic behavior for the solution for particular choices of the heat source, initial, and boundary data are also obtained. A generalization for non-classical moving boundary problems for the heat equation is also given. 2000 AMS Subject Classification: 35C15, 35K55, 45D05, 80A20, 35R35. Keywords: Non-classical heat equation, Nonlinear heat conduction problems, Vol- terra integral equations, Moving boundary problems, Uniform heat source 1. Introduction In this article, we will consider initial and boundary value problems (IBVP), for the one-dimensional non-classical heat equation motivated by some phenomena regarding the design of thermal regulation devices that provides a heater or cooler effect [1-6]. In Section 2, we study the following IBVP (Problem (P1)): u t − u xx = −F ( u x ( 0, t ) , t ) ,0 < x < 1, t > 0 (1:1) u ( 0, t ) = f ( t ) , t > 0 (1:2) ( P1 ) u ( 1, t ) = g ( t ) , t > 0 (1:3) u ( x,0 ) = h ( x ) ,0≤ x ≤ 1 , (1:4) where the unknown function u = u(x,t) denotes the temperature profile for an homo- geneous medium occupying the sp atial region 0 <x<1, the boundary data f and g are rea l functions defined on ℝ + , the initial temperature h(x) is a real function defined on [0,1], and F is a given function of two real variables, which can be related to the evolu- tion of the heat flux u x (0,t) (or of the temperatur e u(0,t)) on the fixed face x =0.In Sections 6 and 7 the source term F is related to the evolution of the temperature u(0,t) when a heat flux u x (0,t) is given on the fixed face x =0. Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 © 2011 Salva et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits u nrestricted us e, distribution, and reproduction in any medium, provided the original work is properly cited. Non-classical problems like (1.1 ) to (1.4 ) are motivated by the modelling of a system of temperature regulation inisotropicmediaandthesourcetermin(1.1)describesa cooling or heating effect depending on the properties of F which are related to the evolution of the heat u x (0,t). It is called the thermostat problem. A heat conduct ion problem of the type (1.1) to (1.4) for a semi-infinit e materi al was analyzed in [5,6], where results on existence, uniqueness and asymptotic behavior for the solution were Problem-Solving Basics for One-Dimensional Kinematics Problem-Solving Basics for One-Dimensional Kinematics Bởi: OpenStaxCollege Problem-solving skills are essential to your success in Physics (credit: scui3asteveo, Flickr) Problem-solving skills are obviously essential to success in a quantitative course in physics More importantly, the ability to apply broad physical principles, usually represented by equations, to specific situations is a very powerful form of knowledge It is much more powerful than memorizing a list of facts Analytical skills and problemsolving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance Such analytical skills are useful both for solving problems in this text and for applying physics in everyday and professional life Problem-Solving Steps While there is no simple step-by-step method that works for every problem, the following general procedures facilitate problem solving and make it more meaningful A certain amount of creativity and insight is required as well Step Examine the situation to determine which physical principles are involved It often helps to draw a simple sketch at the outset You will also need to decide which 1/5 Problem-Solving Basics for One-Dimensional Kinematics direction is positive and note that on your sketch Once you have identified the physical principles, it is much easier to find and apply the equations representing those principles Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities Without a conceptual understanding of a problem, a numerical solution is meaningless Step Make a list of what is given or can be inferred from the problem as stated (identify the knowns) Many problems are stated very succinctly and require some inspection to determine what is known A sketch can also be very useful at this point Formally identifying the knowns is of particular importance in applying physics to real-world situations Remember, “stopped” means velocity is zero, and we often can take initial time and position as zero Step Identify exactly what needs to be determined in the problem (identify the unknowns) In complex problems, especially, it is not always obvious what needs to be found or in what sequence Making a list can help Step Find an equation or set of equations that can help you solve the problem Your list of knowns and unknowns can help here It is easiest if you can find equations that contain only one unknown—that is, all of the other variables are known, so you can easily solve for the unknown If the equation contains more than one unknown, then an additional equation is needed to solve the problem In some problems, several unknowns must be determined to get at the one needed most In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations You may have to use two (or more) different equations to get the final answer Step Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units This step produces the numerical answer; it also provides a check on units that can help you find errors If the units of the answer are incorrect, then an error has been made However, be warned that correct units not guarantee that the numerical part of the answer is also correct 2/5 Problem-Solving Basics for One-Dimensional Kinematics Step Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important—the goal of physics is to accurately describe nature To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units Your judgment will improve as you solve more and more physics problems, and it will become possible for you to make finer and finer judgments regarding whether nature is adequately described by the answer to a problem This step brings the problem back to its conceptual meaning If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to mechanically solve a problem When solving problems, we often perform these steps in different order, and we also tend to several steps simultaneously There is no rigid procedure that will work every time Creativity and insight grow with experience, and the basics of problem solving become almost automatic One way to get practice is to work out the text’s examples for yourself as you read Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and progressing to the more difficult Once you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text Unreasonable Results Physics must ...Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 796065, 21 pages doi:10.1155/2010/796065 Research Article One-Dimensional Compressible Viscous Micropolar Fluid Model: Stabilization of the Solution for the Cauchy Problem Nermina Mujakovi ´ c Department of Mathematics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia Correspondence should be addressed to Nermina Mujakovi ´ c, mujakovic@inet.hr Received 8 November 2009; Revised 24 May 2010; Accepted 1 June 2010 Academic Editor: Salim Messaoudi Copyright q 2010 Nermina Mujakovi ´ c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the Cauchy problem for nonstationary 1D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamical sense perfect and polytropic. This problem has a unique generalized solution on R×0,T for each T>0. Supposing that the initial functions are small perturbations of the constants we derive a priori estimates for the solution independent of T, which we use in proving of the stabilization of the solution. 1. Introduction In this paper we consider the Cauchy problem for nonstationary 1D flow of a compressible viscous and heat-conducting micropolar fluid. It is assumed that the fluid is thermody- namically perfect and polytropic. The same model has been considered in 1, 2, where the global-in-time existence and uniqueness for the generalized solution of the problem on R×0,T,T>0, are proved. Using the results from 1, 3 we can also easily conclude that the mass density and temperature are strictly positive. Stabilization of the solution of the Cauchy problem for the classical fluid where microrotation is equal to zero has been considered in 4, 5.In4 was analyzed the H ¨ older continuous solution. In 5 is considered the special case of our problem. We use here some ideas of Kanel’ 4 andtheresultsfrom1, 5 as well. Assuming that the initial functions are small perturbations of the constants, we first derive a priori estimates for the solution independent of T. In the second part of the work we analyze the behavior of the solution as T →∞. In the last part we prove that the solution of our problem converges uniformly on R to a stationary one. 2 Boundary Value Problems The case of nonhomogeneous boundary conditions for velocity and microrotation which is called in gas dynamics “problem on piston” is considered in 6. 2. Statement of the Problem and the Main Result Let ρ, v, ω,andθ denote, respectively, t he mass density, velocity, microrotation velocity, and temperature of the fluid in the Lagrangean description. The problem which we consider has the formulation as follows 1: ∂ρ ∂t  ρ 2 ∂v ∂x  0, 2.1 ∂v ∂t  ∂ ∂x  ρ ∂v ∂x  − K ∂ ∂x  ρθ  , 2.2 ρ ∂ω ∂t  A  ρ ∂ ∂x  ρ ∂ω ∂x  − ω  , 2.3 ρ ∂θ ∂t  −Kρ 2 θ ∂v ∂x  ρ 2  ∂v ∂x  2  ρ 2  ∂ω ∂x  2  ω 2  Dρ ∂ ∂x  ρ ∂θ ∂x  2.4 in R × R  , where K, A,andD are positive constants. Equations 2.1–2.4 are, respectively, local forms of the conservation laws for the mass, momentum, momentum June 11, 1997 1 PELLPACK: A Problem Solving Environment for PDE Based Applications on Multicomputer Platforms E. N. Houstis, J. R. Rice, S. Weerawarana, A. C. Catlin, P. Papachiou, K Y. Wang and M. Gaitatzes ABSTRACT This paper presents the software architecture and implementation of the problem solving environment (PSE) PELLPACK for modeling physical objects described by partial differ- ential equations (PDEs). The scope of this PSE is broad as PELLPACK incorporates many PDE solving systems and some of these, in turn, include several specific PDE solving methods. Its coverage for 1-D, 2-D and 3-D elliptic or parabolic problems is quite broad, and it handles some hyperbolic problems. Since a PSE should provide complete support for the problem solving process, PELLPACK also contains a large amount of code to sup- port graphical user interfaces, analytic tools, user help, domain or mesh partitioning, machine and data selection, visualization, and various other tasks. Its total size is well over 1 million lines of code. Its open-ended software architecture consists of several software layers. The top layer is an interactive graphical interface for specifying the PDE model and its solution framework. This interface saves the results of the user specification in the form of a very high level PDE language which is an alternative interface to the PELL- PACK system. This language also allows a user to specify the PDE problem and its solu- tion framework textually in a natural form. The PELLPACK language preprocessor generates a Fortran control program with the interfaces, calls to specified components and libraries of the PDE solution framework, and functions defining the PDE problem. The PELLPACK program execution is supported by a high level tool where the virtual parallel system is defined, where the execution mode, file system, and hardware resources are selected, and where the compilation, loading, and execution are controlled. Finally, the PELLPACK PSE integrates several PDE libraries and PDE systems available in the public domain. The system employs several parallel reuse methodologies based on the decompo- sition of discrete geometric data to map sparse PDE computations to parallel machines. An instance of the system is available as a Web server (WebPELLPACK) for public use at the http://pellpack.cs.purdue.edu. keywords: domain decomposition, expert systems, framework, knowledge bases, parallel reuse methodologies, parallel solvers, problem solving environments, programming-in- the-large, programming frameworks, software bus. 1. INTRODUCTION The concept of a mathematical software library was introduced in the early 70s [41] to support the reuse of high qual- ity software. In addition, special journals, conferences, public domain software repositories (e.g., ACM, Netlib), and commercial libraries (i.e., IMSL, NAG) have been established to support this concept. Similar efforts can be found in engineering software, particularly in the areas of structural and fluid mechanics. The increasing number, size, and complexity of mathematical software libraries necessitated the development of a classification and indexing of exist- ing and future software modules. This software is currently organized in terms of the mathematical models involved. A significant effort in this direction is the GAMS on-line catalog and advisory system [5] which has become a stan- dard framework for indexing mathematical software. Information about engineering software can be found in several handbooks which usually describe the applicability and functionality of existing packages. The advances in desktop RESEARCH ARTICLE Open Access Planning and problem-solving training for patients with schizophrenia: a randomized controlled trial Katlehn Rodewald 1 , Mirjam Rentrop 1 , Daniel V Holt 2 , Daniela Roesch-Ely 1 , Matthias Backenstraß 3 , Joachim Funke 2 , Matthias Weisbrod 1,4 and Stefan Kaiser 5* Abstract Background: The purpose of this study was to assess whether planning and problem-solving training is more effective in improving functional capacity in patients with schizophrenia than a training program addressing basic cognitive functions. Methods: Eighty-nine patients with schizophrenia were randomly assigned either to a computer assisted training of planning and problem-solving or a trainin g of basic cognition. Outcome variables included planning and problem-solving ability as well as functional capacity, which represents a proxy measure for functional outcome. Results: Planning and problem-solving training improved one measure of planning and problem-solving more strongly than basic cognition training, while two other measure s of planning did not show a differential effect. Participants in both groups improved over time in functional capacity. There was no differential effect of the interventions on functional capacity. Conclusion: A differential effect of targeting specific cognitive functions on functional capacity could not be established. Small differences on cognitive outcome variables indicate a potential for differential effects. This will have to be addressed in further research including longer treatment programs and other settings. Trial registration: ClinicalTrials.gov NCT00507988 Background Cognitive deficits are important predictors of functional outcome in patients with schizophrenia [1,2]. This find- ing h as motivated the development of different psycho- logical treatment approaches to improve cognitive deficits, which have been subsumed under the term cog- nitive remediation [3]. There is now converging evi- dence that cognitive remediation has moderate effects on cognitive performance [4]. Importantly, these improvements can generalize to functional outcome, particularly when cognitive remediation is combined with comprehensive rehabilitation, such as vocational therapy (e.g. [5-8]). Cognitive remediation covers a broad range of inter- ventions that are het erogeneous with respect to a number of parameters. Importantly, there is consider- able variation in the cognitive functions targeted in training programs. The dominant research focus in the 1980 s and 1990 s was on training procedures addres- sing a particular construct or even a specific task. Most prominently this i ncluded sustained attention based on findings in the Continuous Performance Test and executive function based on Wisconsin Card Sorting Test performance [9,10]. These studies were mostly focused on t he question whether cognitive deficits c an be remediated through training. Recently, more compre- hensive training packages addressing a set of target functions have dominated the literature (e.g. [5,11]). This goes along with a shift in outcome measures. After many of the earlier studies sought to demonstrate improvement on the task trained, a broader effect on neuropsychological test perform ance has subsequently been considered a condition for improvement of patient * Correspondence: stefan.kaiser@puk.zh.ch 5 Psychiatric University Hospital Zurich, Switzerland Full list of author information is available at the end of the article Rodewald et al. BMC Psychiatry 2011, 11:73 http://www.biomedcentral.com/1471-244X/11/73 © 2011 Rodewald et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the ter ms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any me dium, provided the original work is properly cited. relevant outcomes [12]. There is also a growing consen- sus that trials aimed at improving KỸ NĂNG GIẢI QUYẾT VẤN ĐỀ HIỆU QUẢ Prepare by Bui Viet Trung 1 Chủ đề 1: TỔNG QUAN VỀ VẤN ĐỀ VÀ GIẢI QUYẾT VẤN ĐỀ Prepare by Bui Viet Trung 2 Chủ đề 1: TỔNG QUAN VỀ VẤN ĐỀ VÀ GIẢI QUYẾT VẤN ĐỀ 1. Khái niệm “vấn đề”. 2. Các loại vấn đề. 3. Qui trình giải quyết vấn đề. Prepare by Bui Viet Trung 3 1. Khái niệm “vấn đề” • Bạn có chắc chắn là đang thực sự tồn tại một vấn đề? • Vấn đề đó có đáng để giải quyết không? Prepare by Bui Viet Trung 4 Để giải quyết vấn đề cần chú ý: Khi đối diện với một vấn đề sẽ tốt hơn nếu bạn đặt cho mình các câu hỏi: “Chuyện gì sẽ xảy ra nếu… ” hoặc “Giả sử như việc này không thực hiện được……” 1. Khái niệm “vấn đề” Prepare by Bui Viet Trung 5  Có phải vấn đề không?  Không => Không bận tâm.  Có => có phải của ta không?  Không => chuyển sang người khác.  Có => tiến hành phân tích và xữ lý vấn đề.  Không nên lãng phí thời gian và sức lực vào việc giải quyết những vấn đề nếu nó:  Có khả năng tự biến mất  Không quan trọng  Sẽ tốt hơn nếu được giải quyết bởi người khác. 1. Khái niệm “vấn đề” VẤN ĐỀ LÀ GÌ? •Là một cái gì đó khó xử lý hoặc khó giải quyết •Là sự khác biệt trạng thái giữa hiện hữu so với mong đợi, tiêu chuẩn, nguyên trạng Prepare by Bui Viet Trung 6 2. Phân loại vấn đề • 2.1. Các vấn đề sai lệch: • Là một việc gì đó xảy ra không theo kế hoạch/ dự định và cần phải có biện pháp điều chỉnh. Ví dụ: – Máy móc bị trục trặc, – Không nhận được nguyên vật liệu, – Trong nhóm có người bị bệnh, – Bế tắc trong công việc hoặc nhân sự,… Prepare by Bui Viet Trung 7 2. Phân loại vấn đề 2.2. Các vấn đề tiềm tàng: • Là các vấn đề có thể nảy sinh trong tương lai và cần đưa ra các biện pháp phòng ngừa. Ví dụ: – Sự đấu tranh giữa các thành viên trong nhóm, – Nhu cầu gia tăng khiến bạn khó lòng đáp ứng nổi, – Số nhân viên bỏ việc tăng, Prepare by Bui Viet Trung 8 2. Phân loại vấn đề 2.3. Các vấn đề hoàn thiện Là các vấn đề liên quan đến việc làm sao để có năng suất cao hơn, để trở nên hiệu quả hơn và thích ứng nhanh hơn trong tương lai. Ví dụ: –Nâng cấp sản phẩm, nhà cửa trang thiết bị hay phương pháp, –Lắp đặt một hệ thống mới, –Trang bị kỹ năng mới cho nhân viên, –Thay đổi các qui trình để đáp ứng những tiêu chuẩn an toàn mới, Prepare by Bui Viet Trung 9 2. Phân loại vấn đề • Những vấn đề có thể hoặc không thể tiên đoán được:  Có phải mọi vấn đề không thể tiên đoán được thật sự đã không thể lường trước được?  Tại sao những vấn đề có thể tiên đoán được đã không được lường trước? Ghi chú: Lên kế hoạch và suy đoán trước là đã có thể ngăn chặn được một vấn đề trở nên nghiêm trọng hơn. Prepare by Bui Viet Trung 10 [...]... vấn đề Thủ pháp có thể được áp dụng trong tất cả các tình huống có vấn đề là gì? Prepare by Bui Viet Trung 11 2 Phân loại vấn đề Thái độ • Có giải pháp cho vấn đề này khơng? • Có đáng nỗ lực để giải quyết vấn đề này khơng? • Tơi chấp nhận trả một cái giá như thế nào để giải quyết việc này? Prepare by Bui Viet Trung 12 3 Qui trình giải quyết vấn đề: Qui trình giải quyết vấn đề chung: 1.Nhận ra vấn đề. .. 1.Nhận ra vấn đề (Vấn đề thực sự là gì-What? Có phải vấn đề khơng?), 2.Xác định chủ sở hữu vấn đề (who), 3.Hiểu vấn đề - Tìm ra ngun nhân (Thu thập dữ liệu, What- điều gì đang xảy ra? When? Why? ... make sense? 4/5 Problem- Solving Basics for One- Dimensional Kinematics Conceptual Questions What information you need in order to choose which equation or equations to use to solve a problem? Explain... units not guarantee that the numerical part of the answer is also correct 2/5 Problem- Solving Basics for One- Dimensional Kinematics Step Check the answer to see if it is reasonable: Does it make... and use the equation below to find the unknown final velocity That is, 3/5 Problem- Solving Basics for One- Dimensional Kinematics v = v0 + at = + (0.40 m/s2)(100 s) = 40 m/s Step Check to see if