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The differential transform method is used to find numerical approximation of solution to a class of certain nonlinear differential algebraic equations. The method is based on Taylor's theorem. Coefficients of the Taylor series are determined by constructing a recurrence relation. To deal with nonlinearity of the problems, the Fa\`{a} di Bruno's formula containing the partial ordinary Bell polynomials is applied within the differential transform to avoid computation of symbolic derivatives. The error estimation results are presented too. Four concrete problems are studied to show efficiency and reliability of the method. The obtained results are compared to other methods.

We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces($IFNS$), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in $IFNS$ follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.

In this article, the approximation properties of the Szasz-Mirakjan type operators are studied for the function of two variables, and the rate of convergence of the bivariate operators is determined in terms of total and partial modulus of continuity. An associated GBS (Generalized Boolean Sum)-form of the bivariate Szasz-Mirakjan type operators are considered for the function of two variables to find an approximation of B-continuous and B-differentiable function in the Bogel's space. Further, the degree of approximation of the GBS type operators is found in terms of mixed modulus of smoothness and functions belonging to the Lipschitz class as well as a pioneering result is obtained in terms of Peetre K-functional. Finally, the rate of convergence of the bivariate Szasz-Mirakjan type operators and the associated GBS type operators are examined through graphical representation for the finite and infinite sum which shows that the rate of convergence of the associated GBS type operators is better than the bivariate Szasz-Mirakjan type operators.

In this paper, we investigate the approximation properties of the summation-integral type operators as defined by Mishra et al. (Boll. Unione Mat. Ital. (2016) 8:297-305) and determine the local results as well as prove the convergence theorem of the defined operators. Further, check the asymptotic behavior of the operators and obtain the asymptotic formula for the operators, moreover, the quantitative means of Voronovskaja type theorem is also discussed for an upper bound of the pointwise convergent, as well as obtain the Gruss Voronovskaya-type theorem. Finally, the graphical representation is given to support the approximation results of the operators.

The main purpose of this paper is to determine the approximations of Durrmeyer modification of Szasz-Mirakjan operators, defined by Mishra et al. (Boll. Unione Mat. Ital. (2016) 8(4):297-305). We estimate the order of approximation of the operators for the functions belonging to the different spaces. Here, the rate of convergence of the said operators is established by means of the function with derivative of the bounded variation. At last, the graphical analysis is discussed to support the approximation results of the operators.

In this paper, we introduce a new type of Szasz-Mirakjan operators, which preserve a^x, a > 1 fixed and x\geq 0. We study uniform convergence of the operators by using some auxiliary results and also error estimation is given. The convergence of said operators are shown and analyzed by graphics, also in the same direction, we find a better rate of convergence than Szasz-Mirakjan operators by analyzing the graphics. Voronovskaya-type theorem is studied and a comparison is shown under a sense of convexity with Szasz- Mirakjan operators. In the last section, a modified sequence is constructed in the space of integral function.

Our main aim is to investigate the approximation properties for the summation integral type operators in a statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of approximation is determined. Also using weight function, the weighted statistical convergence theorem with the help of Korovkin theorem is obtained. The statistical rate of convergence in the terms of modulus of continuity and function belonging to the Lipschitz class is obtained. To support the convergence results of the proposed operators to the function, graphical representations take place and a comparison is shown with Sz\'asz-Mirakjan-Kantorovich operators through examples. The last section deals with, a bivariate extension of the proposed operators to study the rate of convergence for the function of two variables, additionally, the convergence of the bivariate operators is shown graphically.

The present article deals with the local approximation results by means of Lipschitz maximal function, Ditzian-Totik modulus of smoothness and Lipschitz type space having two parameters for the summation-integral type operators defined by Mishra and Yadav (Tbilisi Mathematical Journal. 11(3), (2018), 175-91). Further, we determine the rate of convergence in the term of the with derivative of bounded variation and for the quantitative means of the defined operators, we establish the quantitative Voronovskaya type and Gr$\ddot{\text{u}}$ss type theorems. Moreover, the examples are given with graphical representation to support the main results.

This paper deals with the approximations of Durrmeyer type generalization of Szasz-Mirakjan operators. We establish the direct results, quantitative Voronovskaya type theorem, Gruss type theorem, A-statistical convergence, rate of convergence in terms of the function with derivative of bounded variation. At last, the graphical analysis, comparison study and numerical representations of proposed operators are discussed.

This paper discusses the properties of a modified version of the Stancu variant Sz\'asz-Mirakjan Kantorovich type operators. We determine the order of approximation in terms of the modulus of continuity and second-order of smoothness, and we obtain the rate of convergence using the Lipschitz space. We also establish a relation using the Peetre-$K$ functional for the proposed operators. An asymptotic formula for these operators is also established to understand the asymptotic behavior. Along with these discussions, we give some convergence properties in weighted spaces using the weight function. Moving ahead in weighted spaces, the Quantitative Voronovskaya-type and Gr\"{u}ss Voronovskaya-type theorems are proved using the weighted modulus of continuity. Furthermore, we determine the rate of convergence of the proposed operators in terms of functions with derivatives of bounded variations. Finally, we provide graphical representations and numerical analysis for our theoretical findings.

In this paper, bivariate Szasz-Mirakjan type operators are introduced along with the estimation of its approximation properties and its rate of convergence. Furthermore, to check the asymptotic behavior of the said bivariate operators, the Voronovskaya type theorem is proved and determined the simultaneous approximation of the operators for first-order partial derivative. Finally, the convergence behavior of the bivariate operators are obtained through graphical representation and validated the numerical results with the results of Szasz-Mirakjan operators for the function of two variables.

The most commonly developed inventory models are the classical economic order quantity model, is governed by the integer order differential equations. We want to come out from the traditional thought i.e. classical order inventory model where the memory phenomena are absent. Here, we want to incorporate the memory effect that is based on the fact economic agents remember the history of changes of exogenous and endogenous variables. In this paper, we have proposed and solved a fractional order EOQ model with constant demand rate where the demand is fully backlogged during shortage time. Finally, a numerical example has been illustrated for this model to show the memory dependency of the system. The numerical example clears that for the considered system the profit is maximum in long memory affected system compared to the low memory affected or memory less system.

In this paper we consider generalized Leontief model. We show that under certain condition the generalized Leontief model is solvable by iterative descent method based on infeasible interior point algorithm. We prove the convergence of the method from strictly positive starting point. A numerical example is presented to demonstrate the performance of the algorithm