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The study of fuzzy variational problems has received significant attention over the past decade due to its successful applications in numerous fields, such as image segmentation and optimal control theory. The fuzzy Euler-Lagrange equations provide the necessary optimality conditions for solving fuzzy variational problems explicitly and have been studied under several differentiability conditions. In this paper, we provide a systematic review to recap the history of variational principle in the calculus of variations and compare it with the existing techniques in the fuzzy setting. We begin with the preliminary concepts and definitions of fuzzy theory and scrutinize the Euler-Lagrange's strategy via systematically searched studies concerning fuzzy variational problems to highlight the importance of improving the existing methods. Finally, we set up the main open problems regarding the limitations of the current approaches, shedding light on future directions. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric function F q + 1 q z for q = 2 , 3 , 4 , 5. This is achieved by means of separating the generalized hypergeometric function F q + 1 q z (q = 2 , 3 , 4 , 5) into even and odd components together with the use of several known infinite series involving reciprocals of the central binomial coefficients obtained earlier by Lehmer. [ABSTRACT FROM AUTHOR]

The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions. [ABSTRACT FROM AUTHOR]

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method. [ABSTRACT FROM AUTHOR]

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense. [ABSTRACT FROM AUTHOR]

We develop a method to obtain approximate solutions for nonlinear systems of Volterra integrodifferential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled Volterra integrodifferential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples and results of the present technique have close agreement with approximate solutions which were obtained with the help of Adomian decomposition method (ADM). [ABSTRACT FROM AUTHOR]

Double Laplace transform method was applied to evaluate the exact value of double infinite series. Further we generalize the current existing methods and provide some examples to illustrate and verify that the present method is a more general technique. [ABSTRACT FROM AUTHOR]

We obtain some general inequalities and establish integral inequalities of the majorization type for invex functions. We give applications to relative invex functions. [ABSTRACT FROM AUTHOR]

The main aim of this work is to present a user friendly numerical algorithm based on homotopy perturbation Sumudu transform method for nonlinear fractional partial differential arising in spatial diffusion of biological populations in animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. The homotopy perturbation Sumudu transform method is a combined form of the Sumudu transform method and homotopy perturbation method. The obtained results are compared with Sumudu decomposition method. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally very attractive. [ABSTRACT FROM AUTHOR]

Anewiterative scheme has been constructed for finding minimal solution of a rational matrix equation of the form X + A* X-1 A = I. The new method is inversion-free per computing step. The convergence of the method has been studied and tested via numerical experiments. [ABSTRACT FROM AUTHOR]

We investigate the Krätzel transform on certain class of generalized functions. We propose operations that lead to the construction of desired spaces of generalized functions. The Krätzel transform is extended and some of its properties are obtained. [ABSTRACT FROM AUTHOR]

We make use of the properties of the Sumudu transform to solve nonlinear fractional partial differential equations describing heat-like equation with variable coefficients. The method, namely, homotopy perturbation Sumudu transform method, is the combination of the Sumudu transform and the HPM using He's polynomials. This method is very powerful, and professional techniques for solving different kinds of linear and nonlinear fractional differential equations arising in different fields of science and engineering. [ABSTRACT FROM AUTHOR]

We investigate the modified Mellin transform on certain function space of generalized functions. We first obtain the convolution theorem for the classical and distributional modified Mellin transform. Then we describe the domain and range spaces where the extended modified transform is well defined. Consistency, convolution, analyticity, continuity, and sufficient theorems for the proposed transform have been established. An inversion formula is also obtained and many properties are given. [ABSTRACT FROM AUTHOR]

The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary differential equations. The improved (G'/G)-expansion method is suggested to solve the space and time fractional foam drainage andKdVequations. The obtained results showthat the presentedmethod is effective and appropriate for solving nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. The method does not require any corrected function or any Lagrangemultiplier and it avoids repeated terms in the series solutions compared to the existing decomposition method including the variational iteration method, the Adomian decomposition method, and Homotopy perturbation method. The approximated solutions obtained converge to the exact solution as N tends to infinity. [ABSTRACT FROM AUTHOR]

The present paper emphasizes Jeffery-Hamel flow: fluid flow between two rigid plane walls, where the angle between them is 2 α . A new method called the reproducing kernel Hilbert space method (RKHSM) is briefly introduced. The validity of the reproducing kernel method is set by comparing our results with HAM, DTM, and HPM and numerical results for different values of H,α, and Re. The results show up that the proposed reproducing kernel method can achieve good results in predicting the solutions of such problems. Comparison between obtained results showed that RKHSM is more acceptable and accurate than other methods. This method is very useful and applicable for solving nonlinear problems. [ABSTRACT FROM AUTHOR]

We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian norm for these functions is computed. Moreover, by employing Jack's lemma, we obtain a convex class in the class of functions of bounded turning and relations with other classes are posed. [ABSTRACT FROM AUTHOR]

We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend the concept of homotopy extension property in homotopy theory for topological spaces to its analogical structure in homotopy theory for topological semigroups. In this extension, we also give some results concerning on absolutely retract and its properties. [ABSTRACT FROM AUTHOR]

We study the relationship between Sumudu and Laplace transforms and further make some comparison on the solutions. We provide some counterexamples where if the solution of differential equations exists by Laplace transform, the solution does not necessarily exist by using the Sumudu transform; however, the examples indicate that if the solution of differential equation by Sumudu transform exists then the solution necessarily exists by Laplace transform. [ABSTRACT FROM AUTHOR]

We introduce and study new sequence spaces which arise from the notions of generalized de la Vallée-Poussin means, invariant means, and modulus functions. [ABSTRACT FROM AUTHOR]

The Sumudu transform of certain elementary matrix functions is obtained. These transforms are then used to solve the differential equation of a general linear conservative vibration system, a vibrating system with a special type of viscous damping. [ABSTRACT FROM AUTHOR]

Let F be a distribution in D' and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x) = F(x) * δn(x) for n = 1, 2, … and {δn(x)} is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ(rs-1)((tanhx+)1/r) exists and Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. for r, s = 1, 2, …, where Kk is the integer part of (s - k - 1)/2 and the constants cj,k are defined by the expansion Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., for k = 0, 1, 2, …. Further results are also proved. [ABSTRACT FROM AUTHOR]

Let F be a distribution in D' and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x) = F(x) * δn(x) for n = 1, 2, … and {δn(x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ(s)[(sinh-1x+)r] does not exists. In this study, it is proved that the neutrix composition δ(s)[(sinh-1x+)r] exists and is given by Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., for s = 0, 1, 2, … and r = 1, 2, …, where cs,k,i = (-1)ss![(k - 2i + 1)rs-1 + (k - 2i - 1)rs+r-1]/(2(rs + r - 1)!). Further results are also proved. [ABSTRACT FROM AUTHOR]

We use the concepts of the quasicoincident relation to introduce and investigate some lower separation axioms such as αT0, αT1, αT½, and αT2 as well as the regularity axioms αR0 and αR1. Further we study some of their properties and the relations among them in the general framework of fuzzy topological spaces. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the p-valent analytic functions in the unit disk and satisfy the differential subordinations z(Ip)r, λ(f(z))(j+1)/(p - j)(ɔp(r, λ)f(z))(j) < (a + (aB + (A- B)β)z)/a(1 + Bz), where ɔp(r, λ) is an operator defined by Sălăgean and β is a complex number. Further we define a new related integral operator and also study the Fekete-Szego problem by proving some interesting properties. [ABSTRACT FROM AUTHOR]

We focus on the equity mutual funds offered by threeMalaysian banks, namely Public Bank Berhad, CIMB, andMalayan Banking Berhad. The equity mutual funds or equity trust is grouped into four clusters based on their characteristics and categorized as inferior, stable, good performing, and aggressive funds based on their return rates, variance and treynor index. Based on the cluster analysis, the return rates and variance of clusters are represented as triangular fuzzy numbers in order to reflect the uncertainty of financial market. To find the optimal asset allocation in each cluster we develop a hybrid model of optimization and fuzzy based on return rates, variance. This was done by maximizing the fuzzy return for a tolerable fuzzy risk and minimizing the fuzzy risk for a desirable fuzzy return separately at different confidence levels. [ABSTRACT FROM AUTHOR]

We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the C(ℵ)-fibration property. We give and prove the relationship between the C(ℵ)-fibration property and an approximate fibration property. Furthermore, we study the pullback maps for C(ℵ)-fibrations. [ABSTRACT FROM AUTHOR]

We extend the concept of relaxed α-monotonicity to mixed relaxed α-β-monotonicity. The concept of mixed relaxed α-β-monotonicity is more general than many existing concepts of monotonicities. Finally, we apply this concept and well known KKM-theory to obtain the solution of generalized equilibrium problem. [ABSTRACT FROM AUTHOR]