Explicit Closed-form Solution Research Articles

Adomian decomposition method for first order PDEs with unprescribed data

In this paper we like to explore the full power of Adomian decomposition method (ADM), specially its symbolic capability. We will demonstrate the standard ADM and ADM with integration factor to compute explicit closed form solutions of first order scalar partial differential equations with unprescribed initial conditions, and even with parameters. These features are those numerical methods fail to do. Our examples include linear/nonlinear, constant/variable coefficients and homogeneous/nonhomogeneous equations. The method of characteristics is also tested and compared with these two ADM methods. We conclude that ADM is excellent among all existing methods.

Correction of the existing solutions for hole/crack problems of composite laminates under coupled stretching-bending deformation

For unsymmetric composite laminates under in-plane forces, transverse loads, out-of-plane bending and/or twisting moments, the stretching and bending deformations would be both induced and coupled together. Through Stroh-like formalism for coupled stretching-bending analysis, several analytical solutions for holes/cracks in laminates were presented in the literature. However, due to the improper use of traction-free hole boundary conditions, most of the existing solutions provide wrong solutions and their associated deflections evaluated by numerical integration of slopes have the multi-valued problem. By revising the traction-free hole boundary condition and setting the single-valued requirement for deflection, some existing solutions are corrected in this paper. The considered problems of holes/cracks in laminates include the one (1) under uniform stretching and bending moment, (2) under uniform heat flow in thickness direction, (3) for electro-elastic laminates, and (4) for Green’s functions with concentrated forces and moments in three orthogonal directions. We not only provide the corrected solutions, but also derive the explicit closed-form solutions of deflection for the above mentioned problems. Additionally, with the corrected Green’s functions, a special boundary element method is developed to assist the verification of corrected solutions and to show its accuracy and efficiency on the problems with holes/cracks in laminates.

EOQ-based pricing and customer credit decisions under general supplier payments

This study investigates an inventory model interfaced with finance, marketing, and operations in a supplier-retailer-customer chain by incorporating some relevant and essential facts. The demand pattern is a multiplicative form of the retailer's selling price and credit period. The retailer receives an upstream advance-cash-credit payment plan from the supplier while offering a downstream cash-credit payment to customers. From the retailer's perspective, offering a longer credit period to customers results in more sales volume but a higher default risk than a shorter one. The retailer's objective is to determine the optimal credit period, selling price, and replenishment time simultaneously to maximize total profit. We develop the model to fit in a general framework that includes various payment types (such as advance, cash, credit, advance-cash, advance-credit, and cash-credit) and numerous previous models as special cases. We derive explicit closed-form solutions to optimal selling price, credit period, and cycle time for five possible cases. We then establish several theoretical results and identify which case is optimal under certain conditions. The purpose of this study is to explore pricing, crediting, and lot-sizing strategies under various payment types – advance, cash, and credit. Finally, we use the analysis numerically to investigate the impact of different payment schemes on the retailer profitability.

Geometry parameter method to estimate stress intensity in V-notched plate and a small crack ahead from V-notch apex

Abstract An approximate, universal approach called the geometry parameter method is developed to estimate the stress intensity factor (SIF) and notch SIF (NSIF) according to linear elastic fracture mechanics (LEFM). NSIF solutions for sharp V-notches include correction coefficients for edge and central notches. The obtained results agree well with results for a double and central V-notched rectangular plate. Furthermore, explicit closed-form solutions are obtained using the proposed method for small crack ahead from a V-notch, and a reliable finite element model is described to calculate crack SIFs in V-notches under mixed loadings. The proposed method is robust and convenient for engineering applications.

Fundamental Issues in Magnetoelectric Transducers: Magnetic Field Sensing Versus Wireless Power Transfer Systems

This paper presents theoretical analysis on fundamental performances of two configurations of magnetoelectric transducers that are used the most in practice, longitudinal-longitudinal and longitudinal-transverse modes, with a focus on magnetic field sensing and wireless power transfer applications. Based on equivalent circuit models and two-port network theory, we derive explicit closed-form solutions for the magnetoelecitric voltage coefficient and the power delivered to a load. We reveal that although the former configuration exhibits some advantages of utilizing as a sensing system, there is no difference in the maximum output power obtained from the two methods. The question on how to reach the upper bound on transferred power is also discussed. The findings can be used as a generalized framework to design an optimal system, depending on desired applications.

Theoretical study on the parameter sensitivity over the mechanical states of inflatable membrane antenna

Large size and high precision inflatable membrane antenna is a promising design for space-based structures. To accurately control its mechanical states, such as shape and stress, the parameter sensitivity of an inflatable membrane antenna is theoretically investigated in this study. Based on the developed analytical matrix model, the influence of loading conditions (pressure, pre-stress) and material properties (elastic modulus, Poisson's ratio) is evaluated, and the stability of inflatable membrane antenna is also demonstrated analytically. Using Taylor expansion, an explicit closed-form solution is obtained, which avoids solving the transcendental equations. The solution is verified by corresponding orthogonal experiment technique and numerical simulations. The research shows that the influence of the variation of the factors on the mechanical states of inflatable membrane antenna will not be amplified, and the ranking of the factors' weights is elastic modulus = pressure > Poisson's ratio > pre-stress.

Portfolio Optimization by a Bivariate Functional of the Mean and Variance

We consider the problem of maximization of functional of expected portfolio return and variance portfolio return in its most general form and present an explicit closed-form solution of the optimal portfolio selection. This problem is closely related to expected utility maximization and two-moment decision models. We show that most known risk measures, such as mean–variance, expected shortfall, Sharpe ratio, generalized Sharpe ratio and the recently introduced tail mean variance, are special cases of this functional. The new results essentially generalize previous results by the authors concerning the maximization of combination of expected portfolio return and a function of the variance of portfolio return. Our general mean–variance functional is not restricted to a concave function with a single optimal solution. Thus, we also provide optimal solutions to a fractional programming problem, that is arising in portfolio theory. The obtained analytic solution of the optimization problem allows us to conclude that all the optimization problems corresponding to the general functional have efficient frontiers belonged to the efficient frontier obtained for the mean–variance portfolio.

Size-dependent vibrations of thermally pre/post-buckled FG porous micro-tubes based on modified couple stress theory

Based on the modified couple stress theory, an analytical approach is presented to study vibrations of thermally pre/post-buckled functionally graded (FG) tubes. The case of simply supported FG micro-tubes with uniform distributed porosity surrounded by nonlinear elastic medium in uniform temperature field is analyzed. Thermomechanical properties are functionally graded across the radius of cross-section of the tube by means of a power law function. The traction free boundary conditions on the inner and outer surfaces of the structure are satisfied. Basic formulations are constructed using the higher-order shear deformation tube theory and the von Kármán type of nonlinear strain-displacement relationships. The governing equations of motion are derived using the Hamilton principle. The dimensionless equations of motion are solved analytically by employing the two-step perturbation technique and the Galerkin procedure. An explicit closed-form solution is obtained to analyze the size-dependent linear and nonlinear free vibrations of micro-tubes in thermal environment. The numerical results are verified by comparison with the existing data in literature for tubes without couple stress effect. Novel numerical results are revealed in the parametric studies to show the large amplitude vibration responses of thermally pre/post-buckled FG porous micro-tubes. The effects of microstructural length scale parameter, functionally graded patterns, porosity distribution parameter, nonlinear elastic foundation, temperature dependence, and geometrical properties of the structure are investigated.

A thermo-viscoelastic analytical model for residual stresses and spring-in angles of multilayered thin-walled curved composite parts

Abstract The thermo-viscoelastic effects play an important role in the development of residual stresses and deformations of fiber-reinforced polymer composite parts during curing. In this paper, an analytical model is proposed to consider thermo-viscoelastic effects on residual stresses and spring-in angles of thin-walled curved composite parts during curing. The proposed model is first established to describe the viscoelastic mechanical behavior of an unidirectional curved composite layer based on the viscoelastic constitutive model with N+1 Maxwell elements. The explicit closed-form solution is obtained by a new iterative solution strategy. Then, the analytical model is extended to investigate the cure-behavior of multilayer curved composite parts. The viscoelastic effects on the residual stresses and spring-in due to tangential and radial thermal-chemical expansion are separately considered to aid in understanding of their individual influence. The analytical results show good agreements with experimental data in literature and numerically predicted results by viscoelastic FEA for the curved composite parts with symmetrical ply layup cured on a wound carbon-epoxy tube tool. Further numerical investigation on the composite parts with unsymmetrical ply layups shows that the viscoelastic effects on the spring-in angles are closely related to the ply layups and the thickness of composite parts and have different responses for those due to tangential and radial effects.

The Capacity and Optimal Signaling for Gaussian MIMO Channels Under Interference Constraints

Gaussian MIMO channel under the joint transmit (Tx) and interference power constraints (TPC and IPC) is studied. A closed-form solution for the optimal Tx covariance matrix in the general case is obtained using the KKT-based approach, up to dual variables. A number of more explicit closed-form solutions are given with optimal dual variables, including full-rank and rank-1 (beamforming) cases as well as the case of identical eigenvectors (typical for massive MIMO settings), which differer significantly from the standard water-filling solution. A “whitening” filter is shown to be an important part of optimal precoding under interference constraints. Sufficient and necessary conditions for each constraint to be redundant are given. Capacity scaling with the SNR is shown to be determined by a natural linear-algebraic structure of sub-spaces induced by channel matrices of multiple users. A number of unusual properties of optimal Tx covariance matrix under the joint TPC and IPC are pointed out and a bound on its rank is established. An interplay between the TPC and IPC is studied, including the transition from power-limited to interference-limited regimes as the Tx power increases. While closed-from solutions for optimal dual variables are given in some special cases, an iterative bisection algorithm (IBA) is proposed to find optimal dual variables in the general case and its convergence is proved for some special cases. Numerical experiments illustrate its efficient performance. Bounds for the optimal dual variables are given.

Provisions for bank deposit withdrawals and portfolio selection

The primary economic function of a bank is to redirect funds from savers to borrowers in an efficient manner, while increasing the value of the bank’s asset holdings in absolute terms. Within the regulatory framework of the Basel III accord, banks are required to maintain minimum liquidity to guard against withdrawals/liquidity risks. In this paper, we analyze a continuous-time mean-variance portfolio selection for a bank with stochastic withdrawal provisioning by relating the reserves as a proxy for the assets held by the bank. We then formulate an optimal investment portfolio selection for a banker by constructing a special Riccati equation as a continuous solution to the Hamilton–Jacobi–Bellman (HJB) equation under mean-variance paradigm. We obtain an explicit closed form solution for the optimal investment portfolio as well as the efficient frontier. The aforementioned modeling enables us to formulate a stochastic optimal control problem related to the minimization of the reserve, depository, and intrinsic risk that are associated with the reserve process.

Green's functions for unsymmetric composite laminates with inclusions.

It is known that the stretching and bending deformations will be coupled together for the unsymmetric composite laminates under in-plane force and/or out-of-plane bending moment. Although Green's functions for unsymmetric composite laminates with elliptical elastic inclusions have been obtained by using Stroh-like formalism around 10 years ago, due to the ignoring of inconsistent rigid body movements of matrix and inclusion, the existing solution may lead to displacement discontinuity across the interface between matrix and inclusion. Due to the multi-valued characteristics of complex logarithmic functions appeared in Green's functions, special attention should be made on the proper selection of branch cuts of mapped variables. To solve these problems, in this study, the existing Green's functions are corrected and a simple way to correctly evaluate the mapped complex variable logarithmic functions is suggested. Moreover, to apply the obtained solutions to boundary element method, we also derive the explicit closed-form solution for Green's function of deflection. Since the continuity conditions along the interface have been satisfied in Green's functions, no meshes are required along the interface, which will save a lot of computational time and the results are much more accurate than any other numerical methods.

An explicit closed-form solution for transverse and longitudinal vibration of beam with multi-directional elastic constraints under an arbitrary moving load

Beams with elastic constraints are widely used in dynamic systems in engineering. A general explicit solution is presented here for the vibration of simple span beam with transverse, rotational and axial elastic boundary constraints due to an arbitrary moving load. The Euler-Bernoulli beam theory is adopted, in which the boundary constraints are treated as multi-directional boundary springs. After the modal analyses, the explicit closed-form solutions of transverse and axial vibration of the beam under a constant, sinusoidal and cosinoidal moving loads are obtained, respectively. And the vibration of a beam subjected to an arbitrary moving load is derived by the superposition of Fourier series. The current analytical solution is exact and can be applied in multiple engineering fields to obtain accurate structural vibrations. In numerical examples, the effects of the boundary springs on the natural frequencies, modes, deflection, bending moment and boundary reaction of the beam are studied in details. The effects of the number of terms in Fourier series of arbitrary moving load are also discussed.

Analytical Solution of Gradually Varied Flow Equation in Non-prismatic Channels

Computation of water surface profile in open channel is considered as an essential component of channel design in open-channel hydraulics. In this regard, gradually varied flow (GVF) is enumerated as the most frequently occurring flow regime in artificial open channels. The analytical and/or semi-analytical solutions of GVF have been acquired for some types of open channels with prismatic channel cross section using both Manning’s and Chezy’s resistance equations. In conventional hydraulic textbooks, the interpretation of water surface profile is mainly based on prismatic channels. Indeed, the non-prismatic term in the numerator of the governing equation creates a further obstacle for proper interpretation. In this paper, an explicit closed-form semi-analytical solution is presented for a non-prismatic rectangular channel based on Chezy’s resistance equation using Adomian decomposition method. The developed semi-analytical solution compares well with both analytical and numerical solutions obtained from predictor–corrector method with very fine spatial resolution. The derived semi-analytical solution can be effectively used to conduct sensitivity analysis of pertinent parameters and compute channel discharge for a given water surface profile.

The Group-Theoretical Analysis of Nonlinear Optimal Control Problems with Hamiltonian Formalism

In this study, we pay attention to novel explicit closed-form solutions of optimal control problems in economic growth models described by Hamiltonian formalism by utilizing mathematical approaches based on the theory of Lie groups. For this analysis, the Hamiltonian functions, which are used to define an optimal control problem, are considered in two different types, namely, the current and present value Hamiltonians. Furthermore, the first-order conditions (FOCs) that deal with Pontrygain maximum principle satisfying both Hamiltonian functions are considered. FOCs for optimal control in the problem are studied here to deal with the first-order coupled systems. This study mainly focuses on the analysis of these systems concerning for to the theory of symmetry groups and related analytical approaches. First, Lie point symmetries of the first-order coupled systems are derived, and then by using the relationships between symmetries and Jacobi last multiplier method, the first integrals and corresponding invariant solutions for two different economic models are investigated. Additionally, the solutions of initial-value problems based on the transversality conditions in the optimal control theory of economic growth models are analyzed.

Adaptive trajectory generation based on real-time estimated parameters for impaired aircraft landing

ABSTRACTThis paper is motivated by a need to address the challenge of securing a safe landing after suffering from inflight impairment. In this paper, a new adaptive generalised model predictive static programming (G-MPSP) is developed to generate a safe emergency landing trajectory for impaired aircraft. Utilising the computationally efficient G-MPSP framework, the proposed algorithm enables adaptation of model parameters based on the prediction errors to ensure reasonable guidance performance. Based on the estimated parameters, a feasible landing trajectory is then generated by the flexible finite-horizon G-MPSP with input constraints. The integrated approach features explicit closed-form solutions for both parameter estimation and trajectory generation. Its effectiveness is demonstrated by simulations in the presence of parameter uncertainties and noises and by comparison studies with the non-adaptive G-MPSP.

Further Results on Pseudo‐Maximum Likelihood Estimation and Testing in the Constant Elasticity of Variance Continuous Time Model

Constant elasticity volatility processes have been shown to be useful, for example, to encompass a number of existing models that have closed‐form likelihood functions. In this article, we extend the existing literature in two directions: first we find explicit closed form solutions of the pseudo maximum likelihood estimators (MLEs) by discretizing the diffusion function and we provide their asymptotic theory in the context of the constant elasticity of variance (CEV) model characterized by a general CEV parameter ρ ≥ 0. Second we obtain bias expansions for those pseudo MLEs also in terms of ρ ≥ 0. We provide a general framework since only the cases with ρ = 0 and ρ = 0.5 have been considered in the literature so far. When the time series is not positive almost surely, we need to impose the restriction that ρ is a non‐negative integer.

Optimal sharing rule for a household with a portfolio management problem

We study an intra-household decision process in the Merton financial portfolio problem. This writes as an optimal consumption–investment problem in finite horizon for the case of two separate consumption streams and a shared final wealth, in a linear social welfare setting. We show that the aggregate problem for multiple agents can be linearly separated in multiple optimal single agent problems given an optimal sharing rule of the initial endowment. Consequently, an explicit closed form solution is obtained for each subproblem, and for the household as a whole. We show the impact of asymmetric risk aversion and market price of risk on the sharing rule in a specified setting with mean-reverting price of risk, with numerical illustration.

An improved model for the horizontal dynamic response of piles in saturated soil

An improved model is developed to analyze the horizontal dynamic response of piles in saturated soil. The proposed model retains the simplicity of the plane strain model while accounting for 3-D effects. The vertical coordinate is suppressed by integrating the governing equations over the length of the pile. The three-dimensional model is reduced to a simpler model that is similar to the plane strain model. Explicit closed-form solutions are obtained by introducing potential functions. The proposed model is verified by comparison with more rigorous solutions. Finally, the influence of pile slenderness, permeability of soil, and other parameters on dynamic soil reaction and dynamic response of pile is investigated. It is revealed that the proposed model avoids the drawbacks of the earlier model and is effective in predicting the horizontal dynamic response of the pile.

A semigroup approach to generalized Black-Scholes type equations in incomplete markets

In this paper we will study an option pricing problem in incomplete markets by an analytic point of view. The incompleteness is generated by the presence of a non- traded asset. The aim of this paper is to use the semigroup theory in order to prove existence and uniqueness of solutions to generalized Black-Scholes type equations that are non-linearly associated with the price of European claims written exclusively on non-traded assets. Then, we derive analytic expressions of the solutions. An approximate representation in terms of a generalized Feynman-Kac type formula is derived in cases where an explicit closed form solution is not available. Numerical examples are also given (see Appendix E) where theoretical approximations and numerical tests reveal a remarkable agreement.