As we know, many existing numerical methods for solving nonlinear fractional time equations under propagation suffer from the phenomenon of decreasing convergence order.

  This

 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?.

Nonparametric methods constitute a highly significant > To address this issue, in 2016, Morovati et al. proposed a generalization of Barzilai-Borwein methods for multi-objective optimization problems. Compared to other gradient-based methods, their approach demonstrated notably higher accuracy and speed, which drew significant interest from researchers in this area. More recently, another generalization of the Barzilai-Borwein method has been introduced, in which a specific weight is assigned to each objective function. The aim of assigning these weights is to reduce the adverse effect of conflicts among objectives on the step size reduction. The authors of the respective study compared their proposed method with that of Morovati et al., and their analysis and numerical results state that their proposed approach has substantial superiority. However, in this thesis, this issue is investigated more critically and thoroughly. It is demonstrated that the above-mentioned analysis is based on a form of false convergence for the newly proposed method and an improper comparison between the two approaches. In fact, it is shown that, contrary to the claims in the literature, the method of Morovati et al. has significant advantages over the proposed approach. This superiority is confirmed both on the test problems presented in the prior study and on a much broader set of ‎benchmark‎  

   در اين پژوهش، يك روش نوآورانه براي نهان گزاري تصاوير ديجيتال ارائه شده است كه تركيبي از تبديل كسينوسي گسسته (DCT)، تبديل موجك گسسته سه‌سطحي (3L-DWT) و تجزيه مقدار تكين (SVD) است. اين روش با هدف افزايش امنيت، غيرقابل تشخيص بودن و مقاومت طراحي شده و قابليت استخراج واترمارك بدون نياز به تصوير اصلي (واترمارك‌گذاري كور) را فراهم مي‌كند.مراحل اصلي روش پيشنهادي شامل پيش‌پردازش تصوير واترمارك با استفاده از نقشه آرنولد، اعمال تبديل‌هاي DCT و DWT، و تجزيه SVD است. واترمارك در ضرايب فركانس پايين حوزه تبديل تصوير ميزبان جايگذاري مي‌شود تا مقاومت بيشتري در برابر حملات مختلف داشته باشد.نتايج آزمايش‌ها نشان مي‌دهد كه روش پيشنهادي در برابر حملات مختلف مانند فيلترها، نويز، حملات هندسي و حذف رديف/ستون مقاومت بالايي دارد و عملكرد بهتري نسبت به روش‌هاي موجود از خود نشان مي‌دهد. اين روش همچنين امنيت بالايي را با استفاده از نقشه آرنولد تضمين مي‌كند.روش پيشنهادي غيرقابل تشخيص بودن بهتري را تضمين مي‌كند كه مقدار آن 57.6303 dB است و مقاومت بهبود يافته‌اي در برابر حملات فيلتر، نويز نمك و فلفل (  ) و چرخش نسبت به روش‌هاي پيشرفته موجود ارائه مي‌دهد. براي فيلتر ميانه با اندازه‌هاي پنجره مختلف، مقدار WNC اين روش برابر با 1 است كه بيشتر از روش‌هاي موجود است.اين تحقيق ضمن ارائه يك روش بهبود يافته براي واترمارك‌گذاري تصاوير ديجيتال، پتانسيل كاربرد در حوزه‌هاي مختلفي مانند حقوق ديجيتال، پزشكي و امنيت نظامي را دارد.

  ‎Because ‎of ‎the ‎shortcomings ‎and ‎numerous ‎challenges ‎that scalarization methods face in solving multiobjective optimization problems, there has been a great deal of interest in recent years in the use of nonparametric methods, which are a generalization of iterative methods in singleobjective optimization.‎‎However, less attention has been paid to the study of accelerated versions of these algorithms. in this paper, an accelerated proximal gradient algorithm is studied for solving multiobjective optimization algorithms in which each objective function is the sum of a differentiable convex function and a proper convex function.‎‎This method, also known as the Fast Iterative Shrinkage- Thresholding Algorithm(FISTA), for scalar optimization. ‎‎The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case, which is ineffective in scalar ‎optimization.‎‎Furthermore, an efficient way to solve the subproblem via its dual representation is presented, and the validity of the proposed method is demonstrated through some numerical experiments

در اين پايان نامه مروري بر انواع روش هاي تقريب براي تجزيه مقاديرتكين[1] براي يك ماتريس را خواهيم داشت.تجزيه SVD ،از اساسي ترين تجزيه هاي جبر خطي عددي به شمار مي رود كه روي ماتريس هاي مستطيلي m*n بكار گرفته مي شود كه امروزه در دنياي پردازش تصوير در زمينه ي فشرده سازي كاربرد فراواني دارد.تجزيه مقدار تكين منجر مي شود تا تصوير را با ابعادي كوچكتر ذخيره كنيم به طوريكه اطلاعات اصلي آن همچنان حفظ شوند.براي فشرده سازي تصوير با استفاده از تجزيه SVD از تقريب ماتريس استفاده مي كنيم كه در فصل دوم اين پايان نامه تقريب و فشرده سازي ماتريس را شرح داده ايم.تقريب ماتريس منجر مي شود تا تصوير را با يك مجموعه ي كوچكتر و در عين حال اصلي تر از مقادير تكين مشاهده كنيم كه روش هاي مختلفي براي اين تقريب مانند نمودار Scree،قانون گاتمن_كايزر،آستانه سخت مقادير تكين و روش هاي مبتني بر آنتروپي،تحليل و ارزيابي مي شوند. در پايان نشان مي دهيم كه هركدام از اين روش ها در شرايط خاصي عملكرد بهينه اي دارند و بسته به ويژگي هاي داده ها،انتخاب مناسب يك روش، مي تواند بهينه ترين مقادير منفرد را براي تحليل و پردازش حفظ كند.    [1] Singular Value Decomposition      

يك خانواده مهم از اين اصلاحات روش هاي گراديان با تآخيرمي باشد. روش هاي گراديان با تاخير (GMR) يك روش تكراري غير يكنوا است كه براي حل معادلات خزي بزرگ متقارن و همچنين معين مثبت توسعه يافته است. اين روش تعميمي از روش هاي تندترين شيب و برزيليا-بوروين است.در اين پايان نامه همگرايي R-خطي اين روش اثبات مي شود و همچنين برخي از روش هاي گراديان دوري مورد مطالعه قرار مي گيرد.بررسي طول گام يوان و خواص طيفي روش هاي گراديان از ديگراهداف اين پايان نامه مي باشد.

  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.

     در اين پايان‌ نامه روشي قوي براي افزودن واتر مارك به تصاوير ارائه مي شود كه اصلي‌ترين پايه‌هاي آن شامل تبديل موجك ، تجزيه و تحليل حالت تجربي دو بعدي ، تبديل كسينوسي گسسته ، بهينه‌سازي انبوه ذرات و تجزيه و تحليل مقدار تكين است. در طول فرايند تعبيه،   سطح 2 براي تجزيه تصوير پوششي به زيرباندها استفاده مي‌شود. همچنين،  براي تجزيه تصاوير و علامت‌گذاري استفاده مي‌شود. علاوه بر اين، تجزيه و تحليل   بر روي باند انتخاب شده از  اجرا مي‌شود. درفاز بهينه‌سازي، براي جستجوهاي پيچيده و چند بعدي استفاده مي‌شود. عوامل تعبيه و مقياس‌بندي با كمك يك كليد امنيتي تعبيه مي‌شوند. تصوير واتر مارك از طريق فرايند استخراج به‌دست مي آيد. نتايج آزمايشي نشان مي‌دهند كه تكنيك پيشنهادي نسبت به چندين حمله هندسي(اعمال نويز) و غير هندسي قوي است.

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.

  A PRACTICAL NUMERICAL APPROACH TO SOLVE A FRACTIONAL LOTKA-VOLTERRA POPULATION MODEL WHITH NON-SINGULAR AND SINGULAR KERNELS Thesis Title:

This thesis investigate a numerical of the convection-diffusion type’s fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coe?cient of the ?rst-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic B-splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems.   

  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

‎Gradient‎ family methods are known as one of the most important methods for solving unconstrained optimization problem. ‎S‎pectral gradient methods are an extensions o‏f gradient methods which aims to overcome some of the drawbacks of gradient methods and produce some efficient methods.‎ ‎ ‎ ‎In this thesis‎, the first goal is to study a new family of spectral gradient methods ‎while ‎t‎his family uses a new stepsize which is determined by a convex combination of the long and short Barzilai–Borwein (BB) stepsizes.‎‎‎ It is also shown that each member of this family have some appropriate quasi-Newton properties.‎‎ ‎ In the sequel, the convergence properties of the new algorithm is investigated and it is shown that the new family of methods is R-superlinearly convergent for two-dimensional problems and R-linearly convergent for the any-dimensional cases. ‎‎‎‎‎‎ In the second part of this thesis, ‎some ‎of ‎cyclic ‎gradient ‎methods ‎have ‎been ‎studied ‎and a‎ ‎new cyclic ‎gradient ‎method ‎and ‎its ‎conve‎rgence properties is studied.‎  

  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 study and analyze high-order Runge–Kutta convolution quadrature (RKCQ)methods for obtaining the numerical solution of nonlinear fractional integro-differential equations(FIDEs) with weakly singular kernels. Wefirst study the existence and uniqueness of solutions for the original problem. Then, the convergence and stability results of the RKCQ method are obtained. Finally, some numerical experiments are reported to illustrate the effectiveness of the proposed schemes.

   As an extension of the 2D fractional Fourier transform (FRFT) and a special case of the 2D linear canonical transform (LCT), the gyrator transform was introduced to produce rotations in twisted space/spatial-frequency planes. It is a useful tool in optics, signal processing and image processing. In this thesis, we study discrete gyrator transforms (DGTs) based on the 2D LCT. Taking the advantage of the additivity property of the 2D LCT, we introduce three kinds of DGTs, each of which is a cascade of low-complexity operators. These DGTs have different constraints, characteristics, and properties, and are realized by different computational algorithms. Besides, we introduce a kind of DGT based on the eigenfunctions of the gyrator transform. This DGT is an orthonormal transform, and thus its comprehensive properties, especially the additivity property, make it more useful in many applications. We also develop an efficient computational algorithm to significantly reduce the complexity of this DGT.

  AbstractIn this thesis, two methods for solving the system of nonlinear equations with largedimensions are investigated. The first method is a hybrid conjugate gradient methodbased on the convex combination of Fletcher–Reeves (FR) and Polak–Ribière–Polyak(PRP) parameters. The global convergence of this algorithm is discussed. Numericalresults show the efficiency and accuracy of the method for problems with largedimensions. In the second method, a spectral conjugate gradient method based on theprojection method has been used for systems of nonlinear monotone equations. Also,this method is suitable for solving with large dimensions problemsKeywords:Conjugate gradient method, Convex combination, Self adaptive, Spectral conjugategradient method, Nonlinear monotone equations.

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.  

    ABSTRACT Two transformations of gradient-descent iterative methods for solving unconstrained optimization are proposed. The first transformation is called modification and it is defined using a small enlargement of the step size in various gradient-descent methods. The second transformation is termed as hybridization and it is defined as a composition of gradient-descent methods with the Picard–Mann hybrid iterative process. As a result, several accelerated gradient-descent methods for solving unconstrained optimization problems are presented, investigated theoretically and numerically compared. The proposed methods are globally convergent for uniformly convex functions satisfying certain condition under the assumption that the step size is determined by the backtracking line search. In addition, the convergence on strictly convex quadratic functions is discussed. Numerical comparisons show better behaviour of the proposed methods with respect to some existing methods in view of the Dolan and Moré’s performance profile with respect to all analysed characteristics: number of iterations, the CPU time, and the number of function evaluations.  KEYWORD : Unconstrained Optimization; Gradient-Descent methods; Muiti Step-Size; Convergence; line Search.

  artial differential equations(PDEs) can model Many physical phenomena.\\\\ The Black–Scholes model is one of the most essential models in finincial mathematics, particularly for American options and European options. Since we can not solve this equation analytically, it seems necessary to provide numerical methods. In this thesis, we consider a compact finite difference method for solving the Black-Scholes equation. Furthermore, we investigate its convergence and stability analysis.\\\\ A higher-order compact finite difference method is introduced for generalised Black–Scholes equation. Moreover, stability analysis, consistent analysis, and convergence analysis of the presented method are investigated. In addition, A consistent and stable numerical scheme for solving a nonlinear option pricing model in illiquid markets is introduced. Finally, some numerical experiments are carried out to illustrate the accuracy and efficiency of mentioned schemes. \\\\

A group of stiff problems, their eigenvalues are separated in to two clusters, one contaning the ”stiff” or fast components and one contaning the ”nonstiff” or slow. By using special exponential fitting techniques we develop a h?s. We obtain the size of their stability regions as a function of the order and the fitting condition. We also obtain condition that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments. Key words: Stiff problems; Explicit Runge-Kutta methods; Exponential fitting; Gap in the eigenvalue spectrum.       A group of stiff problems, their eigenvalues are separated in to two clusters, one contaning the ”stiff” or fast components and one contaning the ”nonstiff” or slow. By using special exponential fitting techniques we develop a h?s. We obtain the size of their stability regions as a function of the order and the fitting condition. We also obtain condition that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments. Key words: Stiff problems; Explicit Runge-Kutta methods; Exponential fitting; Gap in the eigenvalue spectrum.    A group of stiff problems, their eigenvalues are separated in to two clusters, one contaning the ”stiff” or fast components and one contaning the ”nonstiff” or slow. By using special exponential fitting techniques we develop a h?s. We obtain the size of their stability regions as a function of the order and the fitting condition. We also obtain condition that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments. Key words: Stiff problems; Explicit Runge-Kutta methods; Exponential fitting; Gap in the eigenvalue spectrum.   

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.

This thesis primarily presents a solution to functional initial value problem based on artical [22] through alternative legendre polynomials. This method turns the problem into an algebraic equations system from which an appropriate numerical approximation is obtained based on Newton method. Convergence analysis is presented in the end. The given unmerical example further certifies the reliability and validity of the method. The second section of the thesis investigates the vibration fractional equation based on artical [17]. The problem is converted to a sylvester algebraic equations system through Jacobi polynomials. Then a proper numerical approximation obtained from the equations system is presented as the solution. The numerical results for some types of Jacobi polynomials including legendre polynomials, chebychev polynomials second type, third kind, forth kind ans Gegen bauer polynomials ara reviwed through tables and charts. Convergence analysis numerical stability analysis is represented in the end. The important numerical example indicates the method’s accuracy.   

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

   {\\textbf{{چكيده}}} \\\\{\\\\ابتدا، براساس مقاله‌ي\\cite{C2}معادلات ديفرانسيل-انتگرال كسري($FDIEs $) را روي يك كلاس از مشتقات توسعه‌يافته($-B $عملگر) تعريف مي‌كنيم سپس آن را معادلات ديفرانسيل-انتگرال كسري توسعه‌يافته($GFDIEs $) نامگذاري مي‌كنيم. روش هم‌مكاني رابراي فرم‌هاي خطي و غير خطي ها گسترش مي‌دهيم. تقريب‌هاي عددي از ايده روش‌هاي هم‌مكانيبراي حل معادلات انتگرال استفاده مي‌كند. از چندجمله‌اي‌هاي لژاندر به منظور تقريبجواب‌هاي در فضاي با بعد متناهي به همراه همگرايي استفاده مي‌شود. برخي از مثال‌هايكه در آن هسته‌ي$ -B $ عملگرها را تغيير مي‌دهيم در انجامتحقيقات عددي در نظر گرفته مي‌شود. در قسمت دوم، براساس مقاله‌ي\\cite{C3}به معرفي يك روش عددي با مرتبه‌ دقت بالا برايحل معادله‌ي كاتانو با مشتق كسري زماني كه اساس روش طيفي گالركين-لژاندر در بعدمكان و روش هم‌مكاني چپيشف در بعد زمان است، مي پردازيم. در اين روش جواب تقريبيبه جواب واقعي مسأله همگراست و از مرتبه $O(N^{-m}M^{\\sigma} +N^{-m}+M^{\\sigma}) $ مي‌باشدكه در آن $ m \\geq 1 $ و $\\sigma > 1 $ . همچنين، به ترتيب $ N $ و $M $ تعداد گره‌ها در بعد مكان و زمان است. خطايبهينه‌ي كراندار قبلي از روش نيمه گسسته و پايدار و همگرايي طرح تمام گسسته   به‌طور كامل مورد بحث قرار مي‌گيرد. نتايج عدديكارايي اين روش را در بعد زمان و مكان تاييد مي‌كند.

Two spectral conjugate gradient methods based on some quasi-newton equation

an independent unified section.  

The data of most real-world optimization problems (OPs) are often not known exactly at the same time the problem is being solved‎. ‎The reasons for data uncertainty contain measurement errors‎, ‎imprecise data‎, ‎future developments‎, ‎environmental conditions‎. ‎Thus‎, ‎using uncertain robust optimization for optimization problems with uncertain data is essential‎. ‎In robust optimization‎, ‎the uncertain parameters are assumed to belong to a set that is known prior‎, ‎and the focus lies on the worst case‎. ‎The goal is to ensure that the solution is feasible and works well in every possible future scenario‎. ‎An uncertain problem can be solved using the scalarization methods (Benson’s method and elastic constraint method) in multi objective optimization‎. ‎This thesis also focuses on a unified approach to characterizing different kinds of multi objective robustness concepts‎. ‎Based on linear and nonlinear scalarization results for several set order relations‎, ‎together with the help of image space analysis‎, ‎some suitable subsets of scalarization image space are introduced to make equivalent characterizations for upper set (lower set‎, ‎set‎, ‎certainly‎, ‎respectively) less ordered robustness for uncertain multi objective optimization problems‎. ‎In the sequel‎, ‎by virtue of scalar robust optimization and using a deterministic robust counterpart‎, ‎a more general form of the robust optimization is considered in which the objective function and constraints contains uncertain data‎. ‎Moreover‎, ‎the relation between uncertain optimization and the image set is analyzed‎. ‎This idea leads to solve a min-max problem‎. ‎Moreover‎, ‎several necessary and sufficient optimality conditions‎, ‎especially saddle point sufficient optimality conditions for scalar robust optimization problems‎, ‎are obtained‎. ‎Finally‎, ‎a simple example for finding a shortest path is included‎.

  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

  In this theisis, we describe an interactive procedural algorithm for convex multi- objective programming based upon the Tchebyche? method, Wierzbicki’s reference point approach, and the procedure of Michalowski and Szapiro. At each iteration, the decision maker (DM) has the option of expressing his or her objective-function aspirations in the form of a reference criterion vector. Also, the DM has the option of expressing minimally acceptable values for each of the objectives in the form of a reservationvector. Baseduponthisinformation, acertainregionisde?nedforexam- ination. In addition, a special set of weights is constructed. Then with the weights, the algorithm of this paper is able to generate a group of e?cient solutions that provides for an overall view of the current iteration’s certain region. By modi?cation of the reference and reservation vectors, one can ‘‘steer” the algorithm at each itera- tion. From a theoretical point of view, we prove that none of the e?cient solutions obtained using this scheme impair any reservation value for convex problems. The behavior of the algorithm is illustrated by means of graphical representations and an illustrative numerical example. we carry out an extension of the MICA method (modi?ed interactive chebyshev algorithm) for non-convex multiobjective programming. This method is based on the Tchebychev method and in the reference point approach. At each iteration, the decision maker (DM) can provide aspiration levels (desirable values for the objec- tive functions) and also, if the DM wishes, reservation levels (level under which the objective function is not considered acceptable). On the basis of this preferential in- formation, a region of the nondominated objective set is de?ned. In the convex case, considering the aspiration vector as a reference point in an achievement scalarizing function and taking a set of weight vectors, the e?cient solutions generated satisfy the reservation levels. In this work, we analyze the non-convex case. The main re- sult of MICA is veri?ed and demonstrated for the non-convex bi-objective case. The MICA method is not veri?ed in general for multiobjective problems with three or more objective functions, which is demonstrated with a counterexample. we concentrate on reference point based methods in multiobjective programming todemonstrate, asmaincontribution, thatthesolutiontoamultiobjectiveoptimiza- tion problem stays unchanged if the reference point is changed to any point on a set de?nedbymeansoftheoriginalreferencepoint,thenondominatedobjectivesolution and some parameters of the ASF. Concretely, this new set of “equivalent reference points” is the convex linear combination of two straight lines, one containing the original reference point and the other a nondominated objective solution, where the slope of both straight lines is given by the inverses of the weights of the ASF. An illustrative example is used to show the results obtained and an empirical model (application with real data) allows us to highlight possible implications.

  Initial value problems and Boundary value problems have important applications in various branches of pure and applied sciences, including astrophysics, structural engineering, optimization, and economics. In some particular situations it is possible to find a general solution of the equation, but in general it is not possible. In most cases, only approximate solutions can be expected. Accordingly, a large number of methods for the numerical solution of IVPs and BVPs have been proposed in literature. In this thesis, firstly, based on Bernoulli polynomials and using spectral methods, an efficient numerical method is proposed to approximate the analytic solution of an initial value problem. Then, using the Birkhoff- Lagrange-collocation method, a numerical algorithm for solving boundary value problems are studied. Finally, the numerical expriements show that the new methods is efficient.

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.

Boundary value problems have important applications in various branches of pure and applied sciences, including astrophysics, structural engineering, optimization, and economics.‎‎In some particular situations itis possible to find a general solution of the equation, but in general it is not possible. In most cases, only approximate solutions can be expected. Accordingly, a large number of methods for the numerical solution of BVPs have been proposed in literature.‎  In this thesis, firstly, initial value problems and boundary value problems and also some methods to expriements numerical solutions of the problem   is studied.  In the sequel, by using bernoulli polynomials and imposing reproducing kernel and least square methids a novel numerical method is proposed for solving boundary value problems.Finally, the numerical expriements show that the new methods is efficient.

  ‎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:

 ارزيابي ويژگي هاي بافت تصاوير به منظور تعيين شكستگي استخوان