اين پايان نامه رويكردي مبتني بر روش هم‌محلي با استفاده از بي-اسپلاين‌هاي درجه پنجم براي حل عددي معادلات كورت‌وگ-دو فريس مرتبه بالا (هفتم) پيشنهاد مي‌شود. گسسته‌سازي درجهت زمان از طريق يك روش تفاضل متناهي انجام شده و براي تقريب در جهت مكان از تابع بي-اسپلاين درجه پنجم، و مشتقات آن استفاده شده است. اين تحقيق در نظر دارد مسائل مقدار مرزي مرتبه هفتم را به دستگاه مرتبه پايين\\/تر (چهارم) تبديل نمايد وسپس با استفاده از تابع بي-اسپلاين درجه پنجم جواب تقريبي بدست آورد، همچنين در اين پژوهش ابتدا طريقه بدست آوردن توابع بي-اسپلاين‌ها را بررسي مي‌شود و سپس چگونگي محاسبه مشتقات اين توابع مورد مطالعه قرار مي گيرد. در نتيجه، معادله به يك دستگاه جبري تبديل مي‌شود كه با استفاده از خواص ماتريس‌ها حل مي‌گردد. پايداري روش پيشنهادي نيز مورد بحث قرار گرفته و مشخص شده است كه به‌طور نامشروط پايدار است. مثال‌هاي عددي حل شده تا دقت و پايداري اين تكنيك را نشان مي‌دهند. خطا با استفاده از نرم‌هاي بي‌نهايت و نرم دو محاسبه شده است. در مقايسه با نتايج پيشين، مشاهدات نشان مي‌دهد كه اين روش نسبت به بسياري از روش‌هاي ديگر موجود در منابع، برتري دارد.   

 در اين پايان نامه يك تقريب جديد براي معادله انتگرو-ديفرانسيل كسري از هردو نوع ولترا و فردهولم در حالت خطي و غيرخطي ايجاد خواهد شد. عليرغم گام هاي مهمي كه در دستيابي به راه حل هاي عددي كارا و نسبتا دقيق در حل معادلات FIDE ها انجام شده است، همچنان شكاف آشكاري براي توسعه يك روش عددي همه كاره و دقيق كه قادر به حل مسائل متنوع FIDE هاي خطي و غيرخطي با عملگر هاي انتگرال باشد، وجود دارد. براي پر كردن اين شكاف، در اين پژوهش از تكنيك هم مكاني بي اسپلاين مكعبي   به عنوان يك رويكرد قوي و سازگار براي حل طيف گسترده اي از معادلات انتگرو-ديفرانسيل كسري در دو نوع خطي و غير خطي با تركيب كردن عملگر هاي انتگرال ولترا و فردهولم پيشنهاد مي شود. اين روش با به كار گيري ويژگي انعطاف پذيري و كارايي محاسباتي خطوط بي اسپلاين مكعبي، با يك تكنيك يكپارچه راه هاي عددي دقيق تري را ارائه مي كند. از لحاظ   حل پذيري (وجود جواب دستگاه بدست آمده ازگسسته سازي مسئله)، تجزيه و تحليل همگرايي و پايداري   مسئله انجام شده، تاييد بيشتري از دقت و قابليت اطمينان روش هم محلي بي اسپلاين مكعبي را بازگو مي كند كه مي تواند بسيار به حل مسائل FIDE ها با پيچيدگي بيشتر كمك كند. به منظور نشان دادن دقت و كارايي روش   پيشنهادي چند مثال عددي آورده شده و با روش­هاي ديگران كه اين مساله را حل نموده اند و در منابع ذكر شده مقايسه شده است.كلمات كل?د?: معادل? انتگرو-د?فرانس?ل ولترا? كسر?،   معادل? انتگرو-د?فرانس?ل فردهولم كسر?، معادل? انتگرو-د?فرانس?ل كسر?،  حساب كسر?،   بي-اسپلاين مكعب?.

  In this research, a numerical method based on B-spline functions has been developed to solve a set of linear and non-singular singular and non-singular fractional boundary value problems. The approximate solution will be determined by discretizing the main problem with the help of B-spline function in uniform grid points. The convergence analysis of the method is investigated through the matrix approach. Linear and non-linear examples are considered to demonstrate the accuracy and efficiency of the method. The proposed method provides a second-order approximation to solve the investigated problem. This method provides much more accurate results at a lower cost. That is, the cubic spline method with a uniform step length has a lower computational cost. Another advantage of this research is that single points do not appear in the derivative of the fraction.

Higher frame rates are very useful for improving medical diagnosis in fast-moving parts of the heart, especiallyin the valves. To this end, we propose a non-polynomial interpolation method for increasing the frame rate in echocardiography. Besides describing the proposed method, we present two additional contributions: (1) we obtain a closed-form solution, which is continuous and infinitely differentiable; (2) we provide an error analysis of the method. The resulting error bound indicates that the interpolation method is reliable. Finally, to show the efficiency of our proposal in temporal super-resolution, i.e., the increase in frame rate, we apply it to three types of datasets, including a 1D signal, a simulated dataset, and B-mode echocardiography images. Our experimental results show that the Mean Squared Error of the proposed method is reduced from 0.6 to 0.3, while having the same computational complexity compared to cubic B-spline. The quantitative results also indicate that, even with lower selection rates, we can reach a high performance reconstruction while the image quality is not degraded significantly.

  In thesis, a high-order numerical scheme based on B-spline functions is devel- oped for solving a class of nonlinear derivative dependent singular boundary value problems. So far, a second-order approximation method has been obtained from the cubic B-spline method. But in this research, a sixth-order approximation method has been obtained from the cubic bispline interpolator to approximate the solution of the equation. Convergence analysis of the method is established through matrix approach. Four nonlinear examples are considered to demonstrate the accuracy and robustness of the method, One of those does not have an exact solution. the computational time of present method is compared with the others methods.

   Abstract This thesis introduces new concepts of quasi efficiency and quasi proper efficiency for multiobjective optimization problems. These concepts reduce to the most important existing concepts of approximate and quasi efficient solutions. Through the use of quasi efficient solutions, a generalized subdifferential of a vector mapping is introduced, which unifies a number of approximate subdifferentials frequently used in optimization. The general subdifferential is connected to the traditional subdifferential of real functions through scalarization. The use of a generalized subdifferential is employed to express optimality conditions for quasi-efficient solutions. Additionally, this thesis provides optimality conditions for multiobjective optimization problems with cone constraints and polyhedral ordering cones, focusing on approximate proper solutions. A first >Key words: Multiobjective optimization, Quasi e?ciency, approximate solutions, Linear scalarization, Nonlinear scalarization, Vector subdi?erential, Coradiant set, Optimality conditions.   

In this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.Keywords: Bagley–Torvikequation; Chebyshev polynomials; Collocation method; Liouville–Caputo derivative

  در

  In this paper, an efficient numerical scheme is constructed for a generalized fractional subdiffusion problem using a newly proposed generalized weighted shifted Grünwald-Letnikov (gWSGL) approximation for the generalized fractional derivative. The solvability, stability and convergence of the numerical scheme are analyzed using the discrete energy method. It is proven that the temporal convergence order is 2 and this is the best result to date. Simulation is further carried out to demonstrate the accuracy of the proposed numerical scheme

In this Thesis, we propose a new numerical scheme for two-dimensional fractional sub-diffusion problems using non-polynomial spline. The solvability, stability and convergence of the proposed method are established using the well known discrete energy methodology. It is shown that the spatial convergence order is at least 4.5 which improves the best result achieved to date. We also carry out simulation to demonstrate the accuracy and efficiency of the proposed scheme and to compare with other methods.  

In this thesis three numerical methods based on cubic ‎and‎ sextic B-spline for numerical solution nonlinear singular boundary value problems considered. First method based on uniform mesh is of second-order and second method based on non-uniform mesh is of forth-order and third method based on sextic B-spline is of seventh-order. In this thesis the proposed methods not only approximate the solution but also approximate this derivatives and error analysis and convergence of these methods have been analyzed. The end eight linear and nonlinear examples are given to show the applicability and performance methods which show the applicability and performance of the proposed methods.‎  

In this thesis, based on W.K. Zahra, S.M. Elkholy and M. Fahmy (2019) [53], anefficient numerical method is introduced for solving the time-fractional Swift–Hohenberg equation in the sense of Riemann–Liouville derivative. Using rational spline function and nonstandard finite difference technique, numericalmethod is introduced for approximations Swift–Hohenberg. Using the Fourierseries, the method is convergent and unconditionally stable. Also, investigatedthe existence and uniqueness of the proposed method. At the end numericalresults are demonstrated to validate the applicability and the theoretical results.

در اين پايان نامه برخي از روش ها از مرتبه دو را براي چهار نوع اسپلاين انتگرالي درجه دو بررسي شده است.ثابت شده است كه اسپلاين انتگرالي درجه دو داراي همگرايي در تقريب مقدار تابع و تقريب مشتقات مرتبه دوم در نقاط ميان بازه اي يكنواخت هستند.و همچنين بي اسپلاين درجه دوم براي درون يابي يك تابع جلو انتگرال با استفاده از مقادير معلوم انتگرال در زير بازه ها به جاي مقادير تابع در گره ها استفاده مي شود .اين درون يابي اسپلاين انتگرالي درجه دو ناميده مي شود

an independent unified section.  

  AbstractIn this thesis, we investigate the existence of two nontrivial solutions in weighted Sobolev spaces, for a class of fourth-order elliptic equations, with assuming that, nonlinear parts is continuous with a quasicritical growth and it’s potential vanish at infinity, byusingthevariationalmethodandMountainPassTheorem. Therefore, we study the existence of two nontrivial solutions for fourth-order elliptic equations by settingtheAmbrosetti-Rabinwitsconditiononnonlinearpartsandsteepedpotential, by utilizing critical point theory, Mountain Pass Theorem and local minimization. Finally, as an application we will make report the similar results and cocentration phenomenona for second elliptic equations with concave and convex nonlinearities. Keyword: Fourth-order elliptic equation, Mixed nonlinearity, Variational method, cocentration phenomenona, concave-convex nonlinearity, quasicritical growth .

  An element a of a ring R is nil-clean, if a = e + b, where e2 = e ? R and b is a nilpotentelement, andthering R iscallednil-cleanifeachofitselementsisnil-clean. In [22], it was proved that, for a commutative ring R and an abelian group G, the group ring R[G] is nil-clean, i? R is nil-clean and G is a 2-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally ?nite 2-group over a nil-clean ring is nil-clean, and that the hypercenter of the group G must be a 2-group if a group ring of G is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, i? the ring is a nil-clean ring and the group is a 2-group.Keywords:Nil-clean ring, nil-clean group ring

   A cubic trigonometric B-spline collocation approach for the numerical solution of fractional sub-diffusion equation is presented in this paper. The approach is based on the usual finite difference scheme to discretize the time derivative while the approximation of the secondorder derivative with respect to space is obtained by the cubic trigonometric B-spline functions with the help of Grünwald–Letnikov discretization of the Riemann–Liouville derivative.

AbstractIn this thesis, we will consider two numerical methods to approximate of solution of theExtended Fisher–Kolmogorov equation. Both methods under study are >methods, At first a nonlinear high-order difference scheme will be described to solve theExtended-Fisher-Kolmogorov equation . Existence and uniqueness conditions of the solutionwill be analyesed, by utilizing the energy method we proved that the convergent order in maximumnorm is two in temporal direction and four in spatial direction. Solving of numericalresults verifed the theoretical results. In addition a second-order three-level linearly implicit finitedifference method will be studied for solving the extended Fisher–Kolmogorov equation inboth 1D and 2D . The existence and uniqueness of the proposed scheme is proved. In additionBy verifying the convergence and stability of the method, proved that method is second-orderconvergent both in time and space variables, and the method is almost unconditionally stable  

This research speaks about three spline subjects, rstly quadratic b-spline which was used to reconstructan approximating function by using three parameters for that, second a trigonometric spline which wasconstructed by a trigonometric functions mainly to build an approximating function as we will see inmuch details and lastly, quintic b-spline which was used to construct an interpolation method, we willsee in detailed explanations how they have been used and how were the nal results found. Also, wehave demonstrated some examples of error analysis estimations and a comparison with other previousworks, to see which one is better and easier, de nitions are provided with theories and methods toexplain every single step in this work, and an overview of the theories of interpolations for those splinesand their applications in numerical analysis. At the end, the researcher wanted to say that it has beenspoken about cubic spline interpolation in details because its the main spline that is used in our currenttime, and the illustrated examples were of Matlab and Mathematica simulation programs.

  Intervallinearprogrammingwasintroducedinordertodealwithlinearprogramming problems with uncertainties that are modelled by ranges admissible values. Basic tasks in interval linear programming such as calculating the optimal value bounds or set of all possible solutions may be comutationally very expensive. However, if some basis stability criterion holds true then the problems becomes much more easy to solve. We introduce a novel kind of robustness in linear programming. A solution x is called robust optimal if for all realizations of the objective function coe?cients and the constraint matrix entries from given interval domains there are aooropriate choices of the right-handside entriesfrom their intervaldomains suchthat x remains optimal. We propose a method to check for robustness of agiven point, and discuss topological properties of the robust optimal solution set. We illustrate applicability of our concept in tra  ortation and nutrition problems. Since note every problem has a robust optimal solution, we introduce also a concept of an approximate robust solution and develop an e?cient method. We discuss the problem of checking whether a given solution is optimal for each realization of interval data. This problem was studied for particular forms of linear programming problems. we extend the results to a general model and simplify the overall approach, Moreover, we i  ect coputational complexity, too. Eventually, we investigate a related optimality concept of semi-strong optimality.

روش هاي گراديان مزدوج يك خانواده مهم براي حل مسائل بهينه سازي نامقيد هستند. در اين روش ها به دليل عدم نياز به استفاده از ماتريس هسي يا تقريب آن، استفاده كم از حافظه ماشين و خواص همگرايي موضعي و سراسري مناسب، به روش هايي بسيار مطلوب براي حل مسائل بهينه سازي نامقيد در مقياس بزرگ تبديل شده اند. به دليل اين­كه اين الگوريتم ها به طور معمول فقط از اطلاعات مشتق مرتبه اول تابع هدف استفاده مي كنند، بنابران ممكن است همگرايي آنها كند باشد.در اين پايان نامه تركيبي از گراديان مزدوج مقياس يافته و روش مجانب متحرك براي حل مسائل بهينه سازي نامقيد غير خطي در مقياس بزرگ ارائه شده است.در اين روش­ها جهت كاهشي مورد استفاده در هر تكرار به وسيله حل زير مسايل جدايي­پذير محدب توليد شده توسط جهات­مجانبي توليد مي­گردد. همچنين در اين پايان نامه با استفاده از روش­هاي ناحيه اطمينان پارامتر­هاي مجانبي جديد و مؤثري تعريف مي­گردند. بررسي خواص همگرايي و عددي روش­ها هدف بعدي اين پايان نامه است.

A numerical scheme for a fractional sub-diffusion problem using parametric quantic spline

پايان نامه ارشد(6واحدي)

پايان نامه ارشد(6واحدي)

  In this thesis ‎we‎ generalize the notion of ‎$n$‎‎-weak module amenability of ‎$A$‎‎ which is a Banach module over another Banach ‎algebra‎ ‎$U$‎‎ ‎with‎ compatible actions to that ‎of‎ ‎‎‎‎‎$‎(\\sigma)$‎-‎‎‎$n$‎‎-‎weak‎ module amenability ‎for‎ ‎$n\\in‎ ‎‎\\mathbb{N}$‎‎‎ ‎and‎‎ ‎$\\sigma\\in‎ ‎‎Hom_{U}(A)‎$‎‏ ‎.‎We also investigate the relation between this new concept of amenability of ‎$A$‎‎ and the quotient Banach algebra ‎$‎‎A/‎J$‎‎ where ‎$J$‎‎ is the closed ideal of ‎$A$‎‎ generated by elements of the ‎form‎ ‎‎‎$‎‎(a.‎‎\\alpha‎)b -‎‎‎ a(\\alpha.‎‎b)$‎‎‎ ‎for‎‎ ‎$a‎,‎b\\in‎ ‎A$‎‎‎ ‎and‎‎ ‎$\\alpha‎‎ ‎\\in‎‎‎ ‎U$‎‎‎‎ ‎.‎As a consequence ,we show that the semigroup ‎algebra‎ ‎‎‎$‎l‎^{1}(S)‎$‎‎ is ‎$‎‎(\\sigma‎)‎$-‎‎‎‎$‎(‎2n+‎‎‎1)‎‎$-‎weakly module ‎amenable‎ as an ‎‎$‎l^{1}(E)‎$‎ -module for each $‎n\\in \\mathbb{N}‎$‎‎‏ ‎and‎ $‎‎‎‎\\sigma ‎\\in ‎‎‎‎Hom_{l^{1}(E)‎}(‎l‎^{1}(S)‎)‎$‎‎‎ where ‎‎‎$‎S‎$‎ is an inverse semigroup with the set of idempotents ‎‎‎$‎E‎$‎

  Let R be a commutative Noetherian local ring and E the minimal injective cogenerator of the category of R-modules. An R-module M is (Matlis) reflexive if the natural evaluation map M ?? HomR(HomR(M, E), E) is an isomorphism. We prove that if S is a multiplicatively closed subset of R and M is an Rsmodule which is reflexive as an R-module, then M is a reflexive Rs-module. The converse holds when S is the complement of the union of finitely many nonminimal primes of R, but fails in general. Let (R, m) denote a local ring with E = ER(R/m) the injective hull of the residue field. Let p ? SpecR denote a prime ideal with dim R/p = 1, and let ER(R/p) be the injective hull of R/p. As the main result we prove that the Matlis dual HomR(ER(R/p, E) is isomorphic to Rcp, the completion of Rp, if and only if R/p is complete. In the case of R a one dimensional domain there is a complete description of Q ?R Rb in terms of the completion Rb. Keywords: Matlis reflexive, Injective hull, Completion, One dimensional domain, Matlis duality, Minimal injective cogenerator

  آنچه بهينه‌سازي را از كاربردهاي عملي دور نموده است ناشي از دو واقعيت عملي مهم است. يكي تنوع اهداف و معيارهاست كه در بيشتر موارد در تضاد و تقابل با يكديگرند و ديگري نادقيق بودن داده‌هاي اوليه است. براي رويارويي با اين چالش مفهوم پايداري در بهينه‌سازي چندهدفه ارائه شد كه مورد بحث اين پايان‌نامه است.  در اين پايان‌نامه، مفهوم پايداري از مسائل تك‌هدفه به مسائل چندهدفه تعميم داده مي‌شود. منظور از جواب پايدار جوابي است كه به‌ازاي تمام سناريوهاي ممكن كارا بماند. همچنين مفهوم پايداري مينماكس مورد مطالعه قرار مي‌گيرد. فرآيند ارائه شده در اين زمينه، تركيبي است از پايداري در مسائل تك‌هدفه و بهينه‌سازي چندهدفه معين. به‌طور خاص در مسائل بهينه‌سازي چندهدفه‌ي خطي، فرمولي براي شعاع پايداري شدني و شرايط بهينگي قابل اجرا براي جواب‌هاي كاراي ضعيف پايدار ارائه مي‌شود. ضمن معرفي جواب‌هاي كاراي ضعيف بسيار پايدار يك فرآيند محاسباتي نيز براي محاسبه‌ي اين جواب‌ها بررسي مي‌شود. سرانجام يك فرآيند بهينه‌سازي دوهدفه براي تقليل مرز كارا و كمك به تصميم‌گيرنده جهت اتخاذ تصميمي ارجح ارائه مي‌شود. در اين راستا، پايداري جواب و پايداري مدل مطرح مي‌شود. الگوريتم‌هاي محاسباتي مربوطه در اين زمينه براساس تجزيه‌ي وجهي است.

مسائل بهينه سازي ضربي دسته ي خاصي از مسائل بهينه سازي سراسري اند. الگوريتم هاي موجودبهينه سازي سراسري براي حل اين مسائل قابل اجرا هستند ولي كارايي پاييني دارند. اگر عوامل ضربدر بهينه سازي مثبت باشند، جواب بهينه ي اين مسائل يك جواب كارا براي يك مسأله ي بهينه سازيچندهدفه ي نظير است. بنابراين به جاي جستجوي سراسري روي كل فضاي شدني كافيست اين جوابرا در مرز كاراي مسأله ي چندهدفه ي مذكور كه يك مجموعه ي كوچكتر است جستجو كرد. در واقع الگوريتم هاي حل مسائل چندهدفه را مي تواند در اين راستا بسيار كارساز باشد. از جمله الگوريتم هايموثر در اين زمينه الگوريتم ارائه شده توسط ارگوت و شائو در سال 201? است [30]. در اين روشكه براي حل مسائل بهينه سازي چندهدفه ي محدب ارائه شده است، با يك روش برش و كران تقريبياز مرز كارا به دست مي آيد. ارگوت و همكاران[31] در سال 201? با تعميم روش تقريب بيرونيبنسون يك روش مشابه مبتني بر دوگان براي حل مسائل بهينه سازي خطي چندهدفه ارائه كردند. دراين پايانامه بر اساس اين روش الگوريتمي براي حل مسائل بهينه سازي خطي چندهدفه ي ضربي ارائهمي شود.

‎The gradient methods family is an important family of existent method for solving unconstrained optimization problems‎. ‎The Barzilai-Borwein gradient‎‎method is one of the most important gradient methods that has low compution and appropriate speed convergence‎. ‎This method has R-superlinear convergence rate for two-dimensional strictly convex quadratic functions‎. ‎In this thesis‎, ‎we present a new convergence analysis for the BB gradient method that indicate this method has R-superlinear convergence rate with rate $\\sqrt{2}$‎. ‎In the second part of the thesis‎, ‎by combining the conjugate gradient methods and the BB method an algorithm useing of step size BB is presented that keeps appropriate properties of both methods‎. ‎the next topic of the thesis analyzes the convergence properties of the above-mentioned method‎.

  ‎Purpose of the thesis is investigating two numerical methods for the solution of nonlinear systems‎. ‎The main advantages of theses methods is the fact they dont need the derivative of the functions‎. ‎Convergence of the method will be investigated and by some numerical examples the efficiency of the theoretical results will be shown‎.

  In thesis propose nonpolinomial spline and Hermit nonpolynomial spline interpolation and present method to determine optimal value of parametrs which generate minimum error in approximation and used of functions interpolation such the Fouer Series and the Hermite nonpolynomial cubic spline and nonpolynomial cobic spline and interpolated functions for example Runge s Phenomenon Numerical simulations are carried out for the analisis of error in cubic spline and nonpolynomial interpolations.   In this thesis non-polynomial spline for the numerical solution of two-point boundary value problems and singularly perturbed boundary value problems are studied.And it is reduced to sixth order of non-polynomial spline that is used for solving boundary value of second order singularly perturbed.in addition to in both groups of problems, errors and convergence are analyzed.The numerical example are given to illustrate the efficiency of proposed methods.

  This thesis is devoted to study the number of limit cycles bifurcated from the periodannulus of two polynomial vector fields, under polynomial perturbative of degree n.The analysis is carried out by estimatry the number of zeroes of the correspondingAbelian integrals and averaged function. Chebeyshev criterion is one of the toolsfor deriving sharp upper bound for the number of zeroes of the Abelian integrals.Moreover, the distribution of the bifurcated limit cycles is also considere: