Formula For Solution Research Articles

General decay of the Cauchy problem for α-evolution models with β-memory term

In this paper, we investigate the general decay rate of the solutions for a class of fractional Laplacian equations given by u tt + ( − Δ ) α u + u − ∫ 0 t g ( t − s ) ( − Δ ) β u ( s ) d s = 0 , in R n , n ≥ 1 , with 0 ≤ β ≤ α and g satisfying g ′ ( t ) ≤ − η ( t ) g ( t ) . We prove the existence of a solution formula then, establish a general decay result which depends on α, β and η. We also present two illustrative examples by the end.

Regulation of the physicochemical properties of nutrient solution in hydroponic system based on the CatBoost model

Too high or low nutrient solution concentrations adversely affect hydroponic vegetables’ growth and yield performance. In addition, the temperature of the solution and the dissolved oxygen (DO) content of the water are critical factors. A real-time regulation model for nutrient solutions based on the CatBoost model was developed to address the issues of low automation and accuracy of nutrient solution configuration and management. The pH, electric conductivity (EC) and DO of the nutrient solution, and the corresponding water temperature of 96 groups of hydroponic lettuce nutrient solutions (KNO3, MgSO4, Ca(NO3)2, KH2PO4, NH4H2PO4, and microfertilizer) were measured using a gradient combination test designed according to the formula of hydroponic lettuce nutrient solution. A prediction model of the nutrient solution index was constructed using a gradient-enhanced decision tree algorithm and the accuracies of the random forest, XGBoost, and CatBoost algorithms were compared. The results show that the root mean square error (RMSE), mean absolute error (MAE) and coefficient of determination (R2) of the simulated and measured values of the EC prediction model established by the CatBoost model were 248.28 µS/cm, 203.19 µS/cm and 0.946, respectively. At the same time, it was found that the hydroponic variable fertilization control system, composed of four modules–data acquisition, data decision, field control, and manual control–could predict the EC and pH value changes of the nutrient solution and timely add compounds for control. The application of nutrients based on the developed system was good and the weight of the cream lettuce (Flandria RZ) grown using the system was more than 200 g, which meets the requirement for high-quality cream lettuce production. Therefore, the CatBoost model system can be used to improve the hydroponic production system of vegetables by managing nutrients effectively.

Stochastic differential formula and solution of control problem

Exact solution of finite-dimensional linear stochastic differential system with control is derived. Its uniqueness up to stochastic modification is proved. In order to achieve this, detailed grounding of existence for stochastic integral over a process with orthogonal increments is provided. Particularly, density of step-functions set in the set of all integrable functions is proved. Integration by parts formula for stochastic integral is derived. Analogue of Fubini theorem is proved in the case, when one measure is linear and another one is a random orthogonal measure. Stochastic differential formula for finite-dimensional integral functional is established. Formulation of Ito’s stochastic differential formula and its comparison with the result is presented. The solution formula for controlled linear stochastic differential system was conditionally used in applied sciences. Its rigorous proof provides the necessary background for further research.

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

Exact solutions for anisotropic beams with arbitrary distributed loads

The general solution of elasticity for plane anisotropic beams with arbitrary constraints at ends and arbitrary normal and tangential distributed loads on surfaces is derived, which consists of internal forces (i.e., bending moment, shearing force, axial force) and their integrals and derivatives of different orders and load-independent polynomial function sequences of longitudinal coordinates. The method for determining the function sequences is established by resolving the governing equation and boundary conditions of the stress function method. For beams with an elastic symmetry plane, a method for directly determining explicit expressions for all terms of function sequences is provided. Particular solutions of examples are solved using general solution formulas, and the results align excellently with existing exact solutions. Finally, the errors in EBT and TBT when applied to beams made of different materials are analysed.

On the full Kostant–Toda hierarchy and its ℓ-banded reductions for the Lie algebras of type A,B and G

Abstract This paper is concerned with the solutions of the full Kostant–Toda (f-KT) hierarchy in the Hessenberg form and their reductions to the ℓ -banded Kostant–Toda ( ℓ -KT) hierarchies for 1 ⩽ ℓ ⩽ n , where the case with ℓ = 1 is the classical tridiagonal Toda hierarchy and the case with ℓ = n is the f-KT hierarchy. Based on the Hessenberg form of the f-KT hierarchy, we study the f-KT hierarchy and the corresponding ℓ -KT hierarchies on simple Lie algebras of type A , B and G as the root space reductions with proper choices of Chevalley systems. Explicit formulas of the polynomial solutions for the τ-functions are also given in terms of the Schur functions and Schur’s Q-functions.

Solutions of the system of dual matrix equation [formula omitted] in two partial orders

In this paper, we consider the solutions of the system of dual matrix equation AXB=B=BXA in P-star partial order and D-star partial order, respectively. The solvability condition are obtained by applying the singular value decomposition (SVD) and the expression of general solutions to the dual matrix equations are provided. The special case in which A=B is discussed and the properties of {1}−dual generalized inverse are used to verify the accuracy of the explicit solution of dual matrix equation BXB=B. Finally, the explicit formula for the unique minimum-norm solution is given.

On the two nonzero boundary problems of the AB system with multiple poles

The Riemann–Hilbert method is used to study AB systems with two nonzero boundary conditions. In order to reconstruct the potentials, spectral analysis of Lax pair and the corresponding Riemann–Hilbert problem are discussed. For these two nonzero boundary problems, we consider the cases where the sectional analytic functions have double and triple poles, respectively. The derived scattering data has multiple zeros or pairs of multiple zeros, from which the relationship between the eigenfunctions at the corresponding poles can be obtained. By solving algebraic equations, we can derive the formulas of N-order solutions in different cases. As an application, we obtain different types of solitons and breathers in the reflectionless case and the dynamic analysis of these solutions is revealed.

Elasticity solutions for functionally graded beams with arbitrary distributed loads

This paper derives the exact general elasticity solution for functionally graded rectangular beams subjected to arbitrary normal and tangential loads and with arbitrary end constraints. The general solution consists of bending moments and their integrals and derivatives, along with load-independent function sequences of the longitudinal coordinate. The method for determining function sequences has been established based on the stress function method. General solution formulas for stresses, strains and displacements have been derived and used to solve explicit special solutions for six cases involving concentrated forces, uniformly loads, and quadratically distributed loads with different displacement constraints scenarios. The results obtained are compared with existing exact solutions and those of Euler–Bernoulli and Timoshenko beams, and the errors of the latter two are analysed.

Riemann–Hilbert approach and soliton solutions for the Lakshmanan–Porsezian–Daniel equation with nonzero boundary conditions

We construct the Riemann–Hilbert problem of the Lakshmanan–Porsezian–Daniel equation with nonzero boundary conditions, and use the Laurent expansion and Taylor series expansion to obtain the exact formulas of the soliton solutions in the case of a higher-order pole and multiple higher-order poles. The dynamic behaviors of a simple pole, a second-order pole and a simple pole plus a second-order pole are demonstrated.

Sociopolitical context and COVID-19 fatality rates in OECD countries: a configurational approach

BackgroundThe effectiveness of crisis response can be influenced by various structural, cultural, and functional aspects within a social system. This study uses a configurational approach to identify combinations of sociopolitical conditions that lead to a high case fatality rate (CFR) of COVID-19 in OECD countries.MethodsA Fuzzy set qualitative comparative analysis (QCA) is conducted on a sample of 38 OECD countries. The outcome to be explained is high COVID-19 CFR. The five potentially causal conditions are level of democracy, state capacity, trust in government, health expenditure per capita, and the median age of population. A comprehensive QCA robustness test protocol is applied, which includes sensitivity ranges, fit-oriented robustness, and case-oriented robustness tests.ResultsNone of the causal conditions in both the presence and negation form were found to be necessary for high or low levels of COVID-19 CFR. Two different combinations of sociopolitical conditions were usually sufficient for the occurrence of a high CFR of COVID-19 in OECD countries. Low state capacity and low trust in government are part of both recipes. The entire solution formula covers 84 percent of the outcome. Some countries have been identified as contradictory cases. The explanations for their COVID-19 CFR require more in-depth case studies.ConclusionsFrom a governance perspective, the weakness of government in effectively implementing policies, and the citizens’ lack of confidence in their government, combined with other structural conditions, serve as barriers to mounting an effective response to COVID-19. These findings can support the idea that the effects of social determinants of COVID-19 outcomes are interconnected and reinforcing.

A Representation Formula for Viscosity Solutions of Nonlocal Hamilton–Jacobi Equations and Applications

A Representation Formula for Viscosity Solutions of Nonlocal Hamilton–Jacobi Equations and Applications

Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications

The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional calculus literature. In this paper, we first construct an operational calculus of the Mikusiński type for the 1st-level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann–Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for the derivation of the closed-form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st-level GFDs.

Promoting Anthocyanin Biosynthesis in Purple Lettuce through Sucrose Supplementation under Nitrogen Limitation

Although nitrogen deficiency and sucrose are linked to anthocyanin synthesis, the potential role of sucrose in regulating anthocyanin biosynthesis under low nitrogen conditions (LN) in purple lettuce (Lactuca sativa L.) remains unclear. We found that adding exogenous sucrose enhanced anthocyanin biosynthesis but significantly inhibited lettuce growth at high concentrations. Optimal results were obtained using 1 mmol/L sucrose in a low-nitrogen nutrient solution (LN + T1). Chlorophyll fluorescence imaging indicated that the addition of exogenous sucrose induced mild stress. Meanwhile, the activities of antioxidant enzymes (SOD, CAT, and POD) and antioxidant capacity were both enhanced. The mild stress activated the antioxidant system, thereby promoting the accumulation of anthocyanins induced by exogenous sucrose. LN + T1 (low nitrogen nutrient solution supplemented with 1 mmol/L sucrose) up-regulated enzyme genes in the biosynthetic pathway of anthocyanins, including phenylalanine ammonia-lyase (PAL), chalcone synthase (CHS), dihydroflavonol reductase (DFR), flavanone 3-hydroxylase (F3H), flavonoid 3′-hydroxylase (F3′H), flavone synthase II (FNSII), and anthocyanidin synthase (ANS). Additionally, various transcription factors such as AP2/ERF, MYB, bHLH, C2H2, NAC, C2C2, HB, MADS, bZIP, and WRKY were found to be up-regulated. This study elucidates the regulatory mechanism of anthocyanin metabolism in response to the addition of exogenous sucrose under low nitrogen conditions and provides a nutrient solution formula to enhance anthocyanin content in modern, high-quality agricultural cultivation.

The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ 2 + h 2) in L ∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ 4 + h 4) in L ∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.

Numerical methods base on trinomial trees for option pricing

This paper investigates the approach to pricing European options, starting with one-step binomial tree pricing (a relatively simple way to calculate option value). In the next step, an additional possible rate of change of stock price is added to make the model more realistic, resulting in the one-step trinomial tree model. The model bounds the option price under the no-arbitrary principle. The paper then analyzes the circumstances under which options have a xed price by completing the market and giving the solution formula of option price through the model. Last, put-call parity is used to prove the rationality of one-step trinomial model so that the model eectively prevents the occurrence of risk-free arbitrage in the market. This helps traders to price options reasonably in the market and maintains the stability of the options market.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Hypercomplex operator calculus for the fractional Helmholtz equation

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems.

Towards void-free copper filling of low-aspect-ratio heat dissipation through holes in packaging substrate with high H2SO4 concentration electroplating system

The void-free filling of high-density low aspect ratio (AR≤5) through holes (THs) in the packaging substrate is beneficial for promoting the integration, reliability, and heat dissipation efficiency of high-power electronic devices. It is a challenge to fill the low ARs THs compactly in direct current (DC) electroplating. The copper filling process relies heavily on the additives’ complex interactions and multiple process parameters. Herein, an ideal convection-sensitive electroplating solution formula — 2.34 M H2SO4, 0.24 M Cu2SO4·5H2O, 30 mg/L Cl−, 9 mg/L SH110, 5 mg/L NTBC, and 200 mg/L PEG — that was appropriate for low ARs THs filling was provided by investigate the electrochemical characteristics of additives. The critical electroplating parameters were optimized by thoroughly examining their influence on filling process, including current density 1.5 ASD, solution flow rate 2.2 L/min, and temperature 25 °C. The process window was enlarged apparently by the combination of optimal parameters for simultaneous void-free filling of THs with a large AR range of 0.95–5, dense crystal structure, and high layer quality could be achieved. Subsequently, the low ARs THs array filled by the optimized technology was implemented in the packaging substrate as the heat dissipation channel, which significantly promoted the heat dissipation efficiency.

Antiperiodic Solutions for Impulsive ω-Weighted ϱ–Hilfer Fractional Differential Inclusions in Banach Spaces

In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, D0,tσ,v,ϱ,ω, of order σ∈(1,2), in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator D0,tσ,v,ϱ,ω, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions.