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?.

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.

  ‎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      

  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.

We study a multi-group SAIRS-type epidemic model with vaccination. the role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission pattern of the disease among the groups in which the population is divided. This is a natural extension of the homogeneous mixing SAIRS model with vaccination studied in Ottaviano et. al(2021) to a network of communities. We provide a global stability analysis for the model. We determine the value of the basic reproduction number R0 and prove that the disease-free equilibrium is globally asymptotically stable if R0<1. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when R0=1. Moreover, if R0>1, the disease-free equilibrium is unstable and a unique equilibrium exists. First, we investigate the local asymptotic stability of the endemic equilibrium and subsequently its global stability, for two variations of the original model. Last, we provide numerical simulations to compare the epidemic spreading on different networks topologies. 

     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    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.   

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

‎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.‎  

   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.

  AbstractOne of the important concepts from the point of view of theory and computation is the con-cept of proper efficiency in multi-objective optimization. On the other hand, in computationalprocesses, we usually obtain approximate solutions. Therefore, it is necessary to study the prop-erties of these types of solutions and approximate solutions to the related scalar problems to beexamined. Based on this, in this thesis, a generalization of the concept of proper efficiency forproblems with an infinite number of objective functions is investigated. It turns out that someresults for ordinary multi-objective problems cannot be generalized to these problems. In addition,some scalarization methods such as weighted sum and the Chebyshev method are presented re-lated to properly efficient solutions to these problems. In addition, a unified method based on thedirectional Pascoletti–Serafini approach is presented to find efficient, properly efficient, and weaklyefficient solutions as well as similar approximate solutions. In the analysis of these solutions, whilepresenting some characterizations, simple and implementable optimality conditions for efficient

  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.

  ‎Let $V$ be an $n$-dimensional vector space over the finite field consisting of $q$ elements‎. ‎The Grassmann graph of $V$‎, ‎denoted by $\\Gamma_k(V)$‎, ‎is a simple graph whose vertex set is the set of all $k$-dimensional su  aces of $V$‎, ‎with <k<n‎ -‎1$ and two distinct vertices are adjacent if their intersection is $k-1$-dimentional su  ace of $V$‎. ‎Denote by $\\Gamma(n‎, ‎k)_q$ the restriction of $\\Gamma_k(V)$ to the set of all non-degenerate linear $[n‎, ‎k]_q$-codes‎. ‎In this thesis‎, ‎we show that if $n$ is sufficiently large then there exists pairs of codes whose distances in the graphs $\\Gamma_k(V)$ and $\\Gamma(n‎, ‎k)_q$ are distinct‎. ‎Also‎, ‎one 0px; margin-right: 0px; text-indent: 0px;">‎Among other results‎, ‎it is shown that the induced subgraph of $\\Gamma_k(V)$ on projective $[n‎, ‎k]_q$-codes is connected and its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph‎. ‎Then we study the graphs of simplex codes‎. ‎Finally‎, ‎we prove that binary simplex codes of dimension $3$ are precisely maximal singular su  aces of a non-degenerate quadratic form‎.

  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. \\\\

   Let T be a triangular algebra over a commutative ring R, _ be an automorphism of T and Z_(T ) be the _-center of T . Suppose that q : T _ T ??! T is an R-bilinear mapping and that Tq : T ??! T is a trace of q. Our aim is to describe the form of Tq satisfying the commuting condition [Tq; x]_ = 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain classical traingular algebras.

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.‎  

   دستگاه معادلات غيرخطي يكي از مسائل مهم و پركاربرد در رياضيات است. روش‌هاي متفاوتي براي حل اين مسائل تاكنون ارائه شده است. از ميان روش‌هاي تكراري براي حل اين مسائل، مي‌توان به روش نيوتون، روش‌هاي شبه نيوتن و نسخه‌هاي تغيير يافته آن‌ها اشاره كرد.يكي از نقاط ضعف مهم اين روش‌ها بخصوص براي مسائل با ابعاد بزرگ، نياز به محاسبه ماتريس ژاكوبي در هر تكرار و حل دستگاه معادلات خطي متناطر است. تلاش براي ارائه روش‌هاي بدون ژاكوبي براي حل دستگاه‌هاي معادلات غيرخطي در سال‌هاي اخير همواره مورد توجه محققان بوده است. در حالات خاص كه دستگاه معادلات داراي خواص ويژه مي‌باشد، الگوريتم‌هاي بسيار موثري معرفي شده‌اند. يكي از اين رده‌هاي خاص، دستگاه معادلات غيرخطي يكنوا مي‌باشد كه روش‌هاي حل متفاوتي براي آن ارائه شده است. يكي از مهمترين رده هاي موجود براي حل اين مسائل، الگوريتم‌هاي   مبتني بر تصوير است كه بواسطه نياز به حافظه كم، در حل دستگاه معادلات غيرخطي مقياس بزرگ يكنوا كاربردهاي زيادي دارند.   هدف اين پايان‌نامه، ارائه دو خانواده جديد از الگوريتم‌هاي بدون مشتق مبتني بر تصوير است كه از جهاتي شبيه جهات گراديان مزدوج سه‌جمله‌ا‌ي استفاده مي كنند جاييكه ثابت مي شود جهات تعريف شده در شرايط كاهش كافي صدق مي كنند. نتايج عددي به دست آمده نشان مي‌دهد كه اين روش‌ها براي حل اين نوع از مسائل موثر و كارا هستند.

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.   

   {\\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.

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

The main point of this thesis is to prove that therer are several serier. 

In this thesis, we give a novel one parameter algebraic functional equation for fractional resolvent families. 

  In this thesis, we develop simple, yet efficient, procedures for sampling approximations of the two-Parameter Poisson-Dirichlet Process and the normalized inverse- Gaussian process. We compare the efficiency of the new approximations to the corresponding stick-breaking approximations of the two-parameter Poisson-Dirichlet Process and the normalized inverse-Gaussian process, in which we demonstrate a substantial improvement.

an independent unified section.  

  A ring R is called left comorphic if, for each a ? R, there exists b ? R such that Ra = l(b)and r(a) = bR. Examples include (von Neumann) regular rings, and Z p n for a prime p and n ? 1.

  In this thesis, using variational methods and Critical Point Theory, we will investigate the existence and the multiplicity of solutions to   boundary value problems, including Neumannand Dirichlet problems, impulsive problems, fractional differential equations, vaiableexponent equations and etc. Moreover, in some of the results, the positivity and non-triviality of the solutiond will be discussed. Regarding the problems under study, we consider the nonlinear parts as continuous or Carathéodory functions. Then we build the related functionals and discuss the multiplicity of their critical points, obtaining multiple solutions which are the already obtained critical points. Finally, we illuminate the obtained results by presenting various examples.

  1) In this research work, the magnetostriction is investigaterd by using Michelson laser interferometer. 2) Magnetostriction is a property of ferromagnetic materials that is defiend as the change in their dimensions due to the reorientation and rotation of the magnetic domains under the influence of an external magnetic field. 3) In fact, magnetostriction with Michelson interferometer uses two mirrors in a Michelson laser interferometer to obtain the interference pattern. 4) In the experimental set- up, on of the mirrors connected to the metal sample rod such as Fe, Al, Cu and Brass alloy, in to a coil, and is shifted by variation in the magnetic field, and this leads to the change in the interference pattern owning to the magnetostrictive effect.

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  

  در اين پژوهش، فرآيند درج واترمارك شامل اعمال روش بهينه سازي هوشمند DE بر روي تصاوير ميزبان و واترمارك براي يافتن مكان مناسب هر بلوك از تصوير واترمارك در تصوير ميزبان است. سپس جهت بازيابي موفق،‌ خروجي برنامه بهينه سازي در تصوير ميزبان تحت حوزه فركانسي جاسازي مي­شود.   همچنين ضرايب مورد استفاده در جاسازي تصاوير به شكل بهينه بدست آمده است تا بيشترين مقدار   R را بدست دهد. در اين روش، يك بهينه سازي چند هدفه با استفاده از الگوريتم تفاضلي انجام شده است كه در آن مقدار   R در مرحله جاسازي براي تصوير واترمارك و در مرحله استخراج براي تصوير واترمارك بازيابي شده، بسيار مناسب است. در فرآيند درج و استخراج واترمارك، تعبيه و آشكارسازي واترمارك مهمترين بخش مي­باشند چرا كه مقاوم بودن طرح واترماركينگ به بخش تعبيه واترمارك مربوط مي­باشد. سپس مقاوم بودن طرح واترماركينگ در بخش نتايج تجربي مورد ارزيابي قرار مي گيرد و در بخش نتايج تجربي تصوير واترمارك شده را تحت حملاتي از قبيل فشرده سازي تصوير، نويز گوسي و غيره مورد آزمايش قرار داده   و صحت درستي وجود واترمارك مورد ارزيابي قرار خواهد گرفت.

 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.  

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.

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

Banachs fixed point is one of the main fields of research in non-linear analysis of analysis, and is the first to be described in the Banach Rescort. This case has been considered by many researchers for its application and its simplicity, which has been generalized in various ways, such as weakening the contraction inequality, weakening the topology of space, and so on.‎‎‎‎‎This thesis consists of three chapters. In the first chapter, the definitions and the necessary theorems are expressed. The second chapter of this thesis is titled as the fixed point for a weak contraction, which deals with the articles of Chric, Zhang, and Sang, which are presented in full metric space. We also consider the Suzuki case, which is an extension of the Banach contraction theorem, and its generalization‎. We also provide examples and applications of these cases, which will make the results clearer. The third chapter includes  F ‎-‎weak contraction and some of its results. Then we will generalize the contraction of the Banach using the functions called the auxiliary functions introduced by Matkawski and known as the  \\ varphi ‎-‎contraction. We also consider a new type of weak mapping called  F ‎‎-‎‎‎contraction introduced by Imovsky. In this chapter we study the contraction of fixed point theorems for the mapping f using the  F ‎‎-contraction and then the applications of the theorems obtained in the fractal theory are presented.

‎Stochastic differential equations ‎(SDEs) ‎and ‎random‎ differential equations ‎(‎RDE‎s)‎ have been used for many years in a wide range of applications‎. ‎Since these equations dont   have exact ‎solutions‎‎, ‎their numerical solutions have attracted lots of researchers‎. ‎Although classical numerical schemes for ‎ordinary‎ differential ‎equations‎‎   can be often used pathwise for RDEs‎, ‎they rarely attain their traditional order‎.A big class of these equations are Affine ‎random‎ differential equations which have many applications‎. ‎New forms of higher order Taylor-like schemes and also some multi-step ‎methods‎ for RDEs are derived for ‎this‎ kind of equations‎.In the following   ‎numerical schemes based on ‎Itô expansion for ‎RDEs that are driven by an Itô diffusion process‎, ‎i.e‎. ‎the solution of an Itô SDE are considered‎. ‎Under some ‎conditions‎ their convergence will be proposed and some numerical examples are included to confirm ‎the ‎theoretical result‎.

In recent years numerical solution of random ordinary di?erential equations has attracted lots of researchers. Inthisthesisat?rstsomelinearmulti-stepmethodsarederived,andthenundersome assumptions their local error order are established. Then pathwise convergence and B-stabilityofthesemethodsareobtained. Inthefollowing,someimplicitschemesare presented for the pathwise simulation of sti? ordinary di?erential equations, specif- ically an implicit averaged Euler scheme and an implicit averaged midpoint scheme will be considered. Convergence and B-stability proofs of these averaged methods are presented and the numerical schemes are tested for some medical example.

  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)

  In this thesis, first the definition and elementary concepts of analysis and fractionalcalculus is discussed.Then, by using a various fixed point theorems, such as Banach, Krasnoseleskii, Nonlinearcontractions, Leray-schauder Nonlinear Alternative and Leray-schauder degree,the existence and uniqueness of the solutions for a fractional boundary value probleminvolving Erdelyi-Kober operator is investigated.At the end, by using the conception of measures of noncompactness and the Darbotheorem, the existence of a solution for a >involving Erdelyi-Kober operator is discussed.

  ‎In ‎this ‎theisi ‎we ‎invsestiyqte ‎on ‎concept ‎of ‎operator‎-‎valued frames.‎‎In ‎fact, ‎operator-‎‎valued ‎(or g‎-‎‎frames) ‎are ‎generalization‎s of frames and fusion frames and have been used in packets encoding, quantum computing, theory of coherent states and mor. In this article , we give a new formula for operator-valued frames for finite dimensional Hilbert spaces. As an application, we derive in a simple manner a recent result of A. Najati conceerning the approximation of g-frames by Parseval ones. we obtain also some results concerning the best approximation of operator-valued frames by its alternate dual, with optimal estimate.

    Egienvalue problem is one of the most problems in applied mathematics. Amongall of egienvalues the smallest and largest egienvalues have some special importance.Researchers proposed, many numercal methods to solve this problem. In this thesisthe problem of the largest egienvalue of a symetric matrix, convert to a unconstrainedoptimization. Now, we can get a new algorithm by appling an efficient algorithmto solve the generated unconstrained problem. In this thesis, using of a BarzilaiBorwein-like method is proposed. The numerical experimets show the new methodis useful and efficient.

  ‎Nonlinear ‎monotone ‎system ‎of ‎equations ‎is ‎one ‎of ‎the ‎most ‎important ‎problems ‎in ‎applied ‎mathematics ‎where ‎arise ‎in ‎various ‎applications ‎such ‎as ‎subproblems ‎in ‎the ‎generalized ‎proximal ‎algorithms ‎with ‎Bregman ‎distance. ‎There ‎are ‎many ‎various ‎methods ‎to ‎solve ‎this ‎problem, ‎such ‎as ‎Newtons ‎method,quasi‎-newton method and modified version of them. The important weakness of these methods especially for large scale problems is the need to calculate Jacobian matrix in each iteration and solving the corresponding ‎system ‎of linear equations.‎‎projection based algorithms are one of the efficient methods for solving the monotone nonlinear system of equations. In this thesis, Motivated by some conjugate gradient method and the ‎pro‎jection ‎technique ‎two ‎new ‎families ‎of ‎projection ‎based ‎methods ‎is ‎provided ‎to ‎solve ‎nonlinear ‎monotone ‎system ‎of ‎equations. ‎The ‎obtained ‎numerical ‎results ‎show ‎that ‎the ‎methods ‎are ‎effective ‎and ‎efficient.‎‎‎‎

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.

  Let G be a simple graph with the vertex set V(G), r be a non-negative integer and . If the induced subgrsph on S is r-regular, then some upper and lower bounds for the sum of the squares of the entries of the principal eigenvector corresponding to S are presented. Moreover a spectral characterization of families of split graphs, involving their spectral radius and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. An edge-coloring of a graph G with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of G are distinct and the sum of the colors of the edges of G is minimum. The edge-chromatic sum af a graph G is the sum of the colors of edges in a sum edge-coloring of G. In this thesis, We give a polynomial time -approximation algorithm for the edge-chromatic sum problem on r-regular graphs for . Among the other results, the N-completeness of the edge-chromatic sum problem is studied for bipartite graphs and regular graphs. Finally, some upper bounds for the edge-chromatic sum of some split graphs are given.     

In this thesis, we study the fractional calculus and fractional differential equations with Hadamard derivatives, and includes the following parts: in the first chapter, some properties, definitions and theorems of fractional calculus, nonlinear Analysis and fixed point theorems to be introduced that will be used to prove our main results. In the second chapter, the existence and unique   of solutions for a system of Hadamard type fractional differential equations is derived from Leray-Schauders and fixed point theorems guards will be examined. In the next chapter, the existence and uniqueness of solutions ‎using Banachs fixed point theorem for fractional impulsive equations with Hadamard derivatives studied is derived. in the end chapter, existence of solutions for fractional differential equations involving the Hadamard derivatives studied is derivedKeywordsFractional differential, Hadamard fractional derivatives‎, ‎Banachs-fixed point theorem‎, Leray- Schauders theorem‎, Existence of solutions ‎

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.

  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.

 ارزيابي ويژگي هاي بافت تصاوير به منظور تعيين شكستگي استخوان