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Purpose: The aim of this paper is to revisit the albedo for uncertainty. The albedo is considered as a fuzzy value due to some realistic reasons which they will be discussed in details. After defining an appropriate uncertain albedo by using fuzzy set theory, the related energy balance model is also redefined as a fuzzy differential equation by using the concept of fuzzy derivative. Design/methodology/approach: The well-known Earth energy balance model is redefined as a fuzzy differential equation by using the concept of fuzzy derivative. Thus, instead of an ordinary differential equation, a fuzzy differential equation arises which it's solution procedure will be discussed in details. Findings: Results indicate that the fuzzy uncertainty for albedo causes more real results after solving the fuzzy energy balance equation. Considering albedo as a fuzzy number is more realistic than considering a single certain number for albedo of a surface. This is due to this fact that the Earth's surface coverage is not crisp and the boundaries of different types of lands are not consistent. The proposed approach of this paper can help us to provide more realistic climate models and construct dynamical models which can model the albedo based on its variability. Originality/value: In this paper, we defined fuzzy energy balance model as a fuzzy differential equation for the first time. We also, considered albedo as a fuzzy number which is another novel approach. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first- and second-order differential equations subject to nonlocal conditions of integral type. Design/methodology/approach – The authors first transform the given nonlocal boundary value problems of first- and second-order differential equations into local boundary value problems of second- and third-order differential equations, respectively. Then a modified Adomian decomposition method is applied, which permits convenient resolution of these equations. Findings – The new technique, as presented in this paper in extending the applicability of the Adomian decomposition method, has been shown to be very efficient for solving nonlocal boundary value problems of linear and nonlinear first- and second-order differential equations subject to nonlocal conditions of integral type. Originality/value – The paper presents a new solution algorithm for the nonlocal boundary value problems of linear and nonlinear first- and second-order differential equations subject to nonlocal conditions of integral type. [ABSTRACT FROM AUTHOR]

Purpose - The purpose of this paper is to find a semi-analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method to find approximate analytical solutions to fractional oscillators. Design/methodology/approach - The modified differential transform method is used to obtain the solutions of the systems. This approach rests on the recently developed modification of the differential transform method. Some examples are given to illustrate the ability and reliability of the modified differential transform method for solving fractional oscillators. Findings - The main conclusion is that the proposed method is a good way for solving such problems. The results are compared with those obtained by the fourth-order Runge-Kutta method. It is shown that the results reveal that the modified differential transform method in many instances gives better results. Originality/value - The paper demostrates that a hybrid method of differential transform method, Laplace transform and Padé approximations provides approximate solutions of the oscillatory systems. [ABSTRACT FROM AUTHOR]

Purpose - The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert space H with a self-adjoint positive definite operator A under the condition |a(s)|<M1/s1/2 and the first order of accuracy difference scheme for the approximate solution of this initial value problem. Design/methodology/approach - The paper considers the stability estimates for the solution of this problem and the stability estimate for the approximate solution of first order of accuracy difference scheme of this problem. Findings - The paper finds the stability for the fractional Schrödinger differential equation and the first order of accuracy difference scheme of that equation by applications to one-dimensional fractional Schrödinger differential equation with nonlocal boundary conditions and multidimensional fractional Schrödinger differential equation with Dirichlet conditions. Originality/value - The paper is a significant work on stability of fractional Schrödinger differential equation and its difference scheme presenting some numerical experiments which resulted from applying obtained theorems on several mixed fractional Schrödinger differential equations. [ABSTRACT FROM AUTHOR]

Purpose - The purpose of this paper is to offer the theory of a parametrical regulation of market economy development, and the results of the theory development and usage. Design/methodology/approach - Theoretical results of the abstract have been obtained by way of applying the theory of ordinary differential equations, geometrical methods in variation tasks and the theory of dynamic systems. These results have been used for solving a number of practical tasks. Findings - The market economy development parametrical regulation theory structure has been offered. The approach to parametrical regulation of a nonlinear dynamic system's development has been suggested. An assumption about the existence of solution to the task of calculus of variations on the choice of the optimum laws of parametrical regulation within the given finite set of algorithms has been set forward. An assumption about the conditions sufficient for the existence of an extremal's bifurcation point of the task of calculus of variations on the choice of the optimum laws of parametrical regulation within the given finite set of algorithms is presented, formulated and proved. Theory application samples have been provided. Research limitations/implications - Future papers would be focused on studies of rigidness of other mathematic models of economic systems. Practical implications - The research findings could be applied to the choice and realization of an effective budget and tax as well as monetary and loan state policy. Originality/value - The market economy development parametrical regulation theory has been offered for consideration for the first time. [ABSTRACT FROM AUTHOR]

Purpose - The purpose of this paper is to offer the theory of a parametrical regulation of market economy development, and the results of the theory development and usage. Design/methodology/approach - Theoretical results of the abstract have been obtained by way of applying the theory of ordinary differential equations, geometrical methods in variation tasks and the theory of dynamic systems. These results have been used for solving a number of practical tasks. Findings - The market economy development parametrical regulation theory structure has been offered. The approach to parametrical regulation of a nonlinear dynamic system's development has been suggested. An assumption about the existence of solution to the task of calculus of variations on the choice of the optimum laws of parametrical regulation within the given finite set of algorithms has been set forward. An assumption about the conditions sufficient for the existence of an extremal's bifurcation point of the task of calculus of variations on the choice of the optimum laws of parametrical regulation within the given finite set of algorithms is presented, formulated and proved. Theory application samples have been provided. Research limitations/implications - Future papers would be focused on studies of rigidness of other mathematic models of economic systems. Practical implications - The research findings could be applied to the choice and realization of an effective budget and tax as well as monetary and loan state policy. Originality/value - The market economy development parametrical regulation theory has been offered for consideration for the first time. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to propose an economic cybernetics model based on the grey differential equation GM(1,N) for China's high-tech industries and provide the necessary support to assist high-tech industries management departments with their policy making. Design/methodology/approach – Based on the principle of grey differential equation GM(1,N), the grey differential equations of five high-tech industries in China are established using the net fixed assets, labor quantity and patent application quantity as cybernetics variables. After the discretization and first-order subtraction reduction to the simultaneous equation of the five grey models, a linear cybernetics model is resulted in. The structure parameters in the cybernetics system show explicit economic significance and can be identified through least square principle. At last, the actual data in 2004-2010 are introduced to empirically analyze the high-tech industrial system in China. Findings – The cybernetics system for China's high-tech industries are stable, observable, and controllable. On the whole, China's high-tech industries show higher output coefficients of the patent application quantity than those of net fixed assets and labor quantity. This suggests that China's industry development mainly depends on technological innovation rather than capital or labor inputs. It is expected that the total output value of China's high-tech industries will grow at an average annual rate of 15 percent in 2011-2015, with contributions of pharmaceuticals, aircraft and spacecraft, electronic and telecommunication equipments, computers and office equipments, medical equipments and meters by 21, 16, 13, 10, and 28 percent, respectively. In addition, pharmaceuticals, as well as medical equipments and meters, present upward proportions in the gross of Chinese high-tech industries significantly. Electronic and telecommunication equipments, plus computers and office equipments exhibit an obvious decreasing proportion. The proportion of the output value of aircraft and spacecraft is basically stable. Practical implications – Empirical analysis results are helpful for related management departments to formulate reasonable industrial policies to keep the sustained and stable development of the high-tech industries in China. Originality/value – Based on the grey differential equation GM(1,N), this research puts forward an economic cybernetics model for the high-tech industries in China. This model is applicable to the economic system with small sample data set. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to propose a new modification of the Adomian decomposition method for resolution of higher-order inhomogeneous nonlinear initial value problems. Design/methodology/approach – First the authors review the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. Next, the advantages of Duan's new algorithms and subroutines for fast generation of the Adomian polynomials to high orders are discussed. Then algorithms are considered for the solution of a sequence of first-, second-, third- and fourth-order inhomogeneous nonlinear initial value problems with constant system coefficients by the new modified recursion scheme in order to derive a systematic algorithm for the general case of higher-order inhomogeneous nonlinear initial value problems. Findings – The authors investigate seven expository examples of inhomogeneous nonlinear initial value problems: the exact solution was known in advance, in order to demonstrate the rapid convergence of the new approach, including first- through sixth-order derivatives and quadratic, cubic, quartic and exponential nonlinear terms in the solution and a sextic nonlinearity in the first-order derivative. The key difference between the various modified recursion schemes is the choice of the initial solution component, using different choices to partition and delay the subsequent parts through the recursion steps. The authors' new approach extends this concept. Originality/value – The new modified decomposition method provides a significant advantage for computing the solution's Taylor expansion series, both systematically and rapidly, as demonstrated in the various expository examples. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to present a method for solving nonlinear integro-differential equations with constant/variable coefficients and with initial/boundary conditions. Design/methodology/approach – The method converts the given problem into a system of nonlinear algebraic equations. By solving this system, the solution is determined. Comparing the methodology with some known techniques shows that the present approach is simple, easy to use and highly accurate. Findings – The proposed technique allows an approximate solution in a series form to be obtained. Test problems are solved to illustrate the pertinent features of the method. The accuracy of the numerical results indicates that the technique is efficient and well suited for solving nonlinear integro-differential equations. Originality/value – The present approach provides a reliable technique which avoids the tedious work needed by the classical techniques. [ABSTRACT FROM AUTHOR]

Purpose - This paper sets out to study a system of fourth-order obstacle boundary value problems associated with obstacle, unilateral and contact problems. Design/methodology/approach - Differential transform method was used to solve the system. Findings - It is demonstrated that the proposed scheme validates for this type of problems. Originality/value - It is the first time, to the best of one's knowledge, that the method is applied to obstacle boundary value problems. Also, the technique implemented in this study can be used for this type of physical length sensitive problems. [ABSTRACT FROM AUTHOR]

Purpose - Initial value problems for the one-dimensional third-order dispersion equations are investigated using the reliable Adomian decomposition method (ADM). Design/methodology/approach - The solutions are obtained in the form of rapidly convergent power series with elegantly computable terms. Findings - It was found that the technique is reliable, powerful and promising. It is easier to implement than the separation of variables method. Modifications of the ADM and the noise terms phenomenon are successfully applied for speeding up the convergence of non-homogeneous equations. Research limitations/implications - The method is restricted to initial value problems in which the space variable fills the whole real axis. Modifications are required to deal with initial boundary value problems. Further, the input initial condition is required to be an infinitely differentiable function and obviously, the convergence radius of the decomposition series depends on the input data. Practical implications - The method was mainly illustrated for linear partial differential equations occuring in water resources research, but the natural extension of the ADM to solving nonlinear problems is extremely useful in nonlinear studies and soliton theory. Originality/value - The study undertaken in this paper provides a reliable approach for solving both linear and nonlinear dispersion equations and new explicit or recursively-based exact solutions are found. [ABSTRACT FROM AUTHOR]