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

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

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

‎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 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.

   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.   

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.

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.

  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.

  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.  

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.

  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.

  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.

  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.

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