For a graph G, the first Zagreb index is defined as the sum of the squares of the vertex degrees, while the second Zagreb index is the sum of the products of the degrees of adjacent vertices. The aim of this paper is to completely characterize n-vertex trees with given k ? 1 vertices that have a fixed maximum degree ? ? 3 with respect to the maximal and minimal Zagreb indices. Furthermore, our results provide detailed insights into the structure of extremal trees and are equally applicable to the class of chemical trees.

  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 مدل شكارچي-شكار با دفاع القايي و بيماري در شكار كه از ديدگاه تكامل زيستي و اكو-اپيدميولوژي ساخته شده است، مورد مطالعه قرار مي‌گيرد. هدف اصلي اين پايان نامه بررسي تأثير بيماري بر پايداري جمعيت در مدل شكارچي-شكار با دفاع القايي است. ابتدا، مثبت بودن و كرانداري يكنواخت جواب‌هاي مدل اشاره شده اثبات مي‌شود. سپس، وجود و پايداري نقاط تعادل، بخصوص نقاط تعادل همزيستي شكار و شكارچي مورد مطالعه قرار مي‌گيرد. در اين مدل حداكثر نه نقطه تعادل وجود دارد. از يك تبديل پارامتري پيچيده براي بررسي ويژگي‌هاي نقاط تعادل همزيستي مدل استفاده مي‌كنيم. همچنين شرايط كافي براي وجود انشعاب هاپف بدست مي‌آيد. براي تحليل انشعاب هاپف و تعيين جهت سيكل‌هاي حدي در مدل شكارچي-شكار با دفاع القايي و بيماري، ضريب اول لياپانوف در مقادير بحراني پارامتر انشعاب محاسبه مي‌شود. در خاتمه شبيه‌سازي‌هاي عددي براي تكميل نتايج تحليلي انجام مي‌شود  

We introduce the notion of coretractable modules. A module $M$ is said to be coretractable if for every nonzero factor module $M/N$ (where $N \\leq M$), there exists a nonzero $R$-homomorphism $f: M/N \\to M$. We prove that all right (left) modules over a ring are coretractable if and only if the ring is Morita equivalent to a finite product of local right and left perfect rings.  

 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?.

Iterative methods for solving multi-objective optimization problems have greater computational complexity compared to single-objective problems. Accordingly, gradient-based methods that do not use higher-order derivatives are more desirable for this purpose. On the other hand, these methods have slower convergence rates. One important idea to address this issue is to use information from previous iterations alongside the gradient of the current iteration to construct the desired direction. The most basic methods based on this idea are conjugate gradient methods. In this regard, this thesis addresses some methods that, by employing suitable parameters and utilizing information from previous iterations, yield relatively fast processes for solving multi-objective optimization problems. While investigating the convergence of these methods, their computational superiority is demonstrated using some standard test problems in multi-objective optimization.  

   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.

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      

  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 thesis, we examine theorems about bifurcation from one-dimensional kernels and generalizations of the Crandell-Rabinowitz theorem. Next, using a bifurcation theorem from one dimensional kernels, the bifurcation in Activator-inhibitor systems are discussed. Also, the effect of nonlocal diffusion on bifurcations and the formation of spatially heterogeneous patterns in the case when the rate of dispersion of the inhibitor is small enough, is study. Unstable steady state solutions and existence of Turing instability for the mentioned nonlocal systems is investigated.

   Abstract    In the last two decades, autonomous vehicles have received a widespread use and attention. In this thesis, navigation of such machines is investigated. Path planning is one of the important part of this navigation. Duo to the long-term use of roads and lack of their maintenance, the roads which autonomous vehicles need to pass have tracks. In these cases, while an autonomous vehicle passes through these cracked areas at high speed, it will increase the sense of bump and even deviate from the originally planned route. Further, this may potentially cause vehicle damage. In this thesis, an adjustable speed navigation method, called Bat-Pigeon algorithms is investigated. First a review on heuristic and meta-heuristic algorithms is presented. After that, Bat-algorithm and Pigeon-algorithm for optimization are studied. Next, an image processing technique is introduced and by using of that process a combination of Bat- and Pigeon- algorithms, that is Bat-Pigeon-algorithm, is investigated for navigation and path planning of autonomous vehicles.       Keywords: Heuristic algorithm, Meta-heuristic algorithm, Swarm-intelligence optimization, Bat-algorithm, Pigeon-algorithm, Autonomous vehicle, Bat-Pigeon algorithm, Path planning, Road track detection, Image processing.   

  در

   تعامل شكار-شكارچي يا منبع-مصرف كننده، اساسي ترين و مهم ترين فرآيند در پويايي جمعيت است. بسياري از گونه   ها، مانند گياهان تك باره و جانوران يك بار زا كه پس از زادآوري مي ميرند، داراي نسل هاي ناهمپوشان گسسته هستند و تولد آنها در فصول توليد مثل به طور منظم اتفاق مي افتد. فعل و انفعالات آنها با معادلات تفاضلي توصيف و يا به صورت نگاشت هاي زمان-گسسته فرموله مي شوند. در اين پايان نامه، انشعابات را در يك مدل شكار-شكارچي گسسته با تابع پاسخ غير يكنوا كه توسط تابع ساده شده هالينگ IV توصيف شده است، مطالعه مي كنيم. همچنين ثابت مي كنيم كه مدل فوق انشعاب هاي مختلفي از هم بعد 1 را نشان مي دهد، كه شامل انشعابات فولد، انشعاب ترا بحراني، انشعاب فليپ و انشعاب نيمارك-ساكر مي باشد، زيرا مقادير پارامترها متفاوت است. علاوه بر اين، وجود انشعاب بوگدانوف -تاكنز از هم بعد 2 را مشخص و عبارات تقريبي منحني هاي انشعاب را محاسبه مي كنيم شبيه سازي هاي عددي نيز دهد براي نشان دادن تحليل نظري ارائه شده اند اين نتايج نشان مي دهد كه انشعاب بوگدانوف-تاكنز از هم بعد 2 در تكيني تبهگن در هر سه نسخه زمان-پيوسته، زمان-گسسته و زمان-تاخيري از مدل شكار-شكارچي با تابع پاسخ غير يكنوا برقرار است.

  In recent years, finding the closest target for the under-evaluation decision making units (DMU) has attracted the attention of researchers significantly, and numerous articles have been published in this field. In some of these articles, the related efficiency measure does not satisfy the strong monotonicity property. Since this property plays a very important role in comparing and ranking units, it is very desirable to present methods that, while finding the closest efficient model, their efficiency measure is strongly monotone. Mainly, researches done in this field are divided into the following two general categories: A) The methods that obtain all of the full dimensional efficient facets or their extended versions and then obtain the distance of the DMU under evaluation to these facets. B) Methods that, instead of obtaining full dimensional efficient facets, using some mixed integer linear programming models, implicitly calculate the distance of the under evaluation DMU to the strongly efficient frontier. In both cases, based on the obtained distance, a well-defined strong efficiency measure is introduced. This thesis investigates these methods in detail using some real numerical results.

   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.  

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, 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.

the main disadavantages of the newton method are that the computation of the hessian matrix is a difficult 

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.   

Diagonal quasi-Newton method 

   {\\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.

  ‎‎‎In multiobjective optimization, several different objective should be optimized which are in conflict in general. Thus, the objective space is a set of vectors. For comparing these vectors a partial order is needed. Almost in all cases, this order is defined by a cone. This thesis studies multiobjective problems in which the related cone is a polyhedral cone. For polyhedral cones, it is shown how to find vectors in the positive dual cone that are needed for a scalarized objective function. Instructive examples are presented. In this thesis, by using a special kind of polyhedral cone namely, dilating cones and applying nonlinear scalarization proper are characterized. A similar characterization is derived for weakly efficient solution, for which no convexity hypotheses are required. Finally, when the feasible set is given by a cone constraint, some necessary and sufficient optimality conditions via a kind of scalar nonlinear Lagrangian are obtained.

  Quasi-newton methods are an important class of iterative methods for solving unconstrained optimization.These methods can be used when the Hessian matrix is difficult or time consuming to eraluate. Insted of using an estimate of the Hessian matrix, these methods buildup an approximate Hessian matrix by using gradient information. This family is one of the popular algorithms and has many advantages, but there are some drawbacks on them. This methods only use the gradient valuse and don't use the function valuse.Also, in most cases, the quasi newton method can't guarantee the positive definite of Hessian matrix.The modified quasi-newton methods suggest some proposals to overcoming this disadvantages.\\\\In this thesis, two new family of modified quasi-newton methods were investigated. firstly, a modified BFGS algorithm with global convergence properties for nonconvex functions were presented. In the second part by modifying the interpolation conditions to approximate the quadratic model of the function, a new modified quasi-newton equation is introduced. Based on this equation, a modified BFGS algorithm is presented.

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  

 arabolic equations are an important class of partial diferential equations which have many applications in science. Since these equations dont have exact solution, Their numerical solutions have at- tracted lots of researchers. A big class of these equations are Burgers equations. In this thesis some numerical schems based on Splitting methods are derived for this kind of equations. Also some Splitting methods with higher orders for solving a wide range of parabolic equation will be investigated.  

 complex linear systems have many applications in scientific computing and applied engineering most of these equations don't have exact solution.therefore the main purpose of this thesis is to obtain some numerical methods for solving these equations.by obtaining the spectral radius of the introduced method,their convergence analysis will beinvestigated.finally by some numerical examples efficiencity of the methods will be explained

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.

  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.

  This thesis concerns the stability and bifurcations in two planer systems of ODE’s,which are models of a prey - predator system and a SIR epidemic model.It is proved that the predator - prey model exhibits several bifurcations.These bifurcations are ecologically important and the saddle - node bifurcation andcodimension 2 Bogdanov - Takens bifurcation especially will lead to the potentiallydramatic variation of the system dynames.In the SIR model, it is proved that under some conditions the system exhibits backward bifurcation and Hopf birfurcation.

  The dynamics of a general di?usive predator–prey system is considered. Existence and nonexistence of non-constant positive steady state solutions are shown to iden- tify the ranges of parameters of spatial pattern formation. Bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non-constant steady state solutions are studied. Keywords: Reactiondi?usion; Patternformation; Globalbifurcation; Predatorprey; StrongAllee e?ect; Spatiotemporal patterns

  ‎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.