Formula For Solution Research Articles

A Note On The Dirichlet-Neumann Problem in The Upper Half Unit Disc

This study addresses the solvability and explicit solutions of various boundary value problems (BVPs) for inhomogeneous equations within the upper half unit disc. Specifically, we investigate the Dirichlet, Neumann, and mixed Dirichlet-Neumann problems for the inhomogeneous Cauchy-Riemann and Bitsadze equations. By employing analytical techniques and function space theory, we establish necessary and sufficient conditions for the existence of solutions. Furthermore, explicit solution formulas are derived under these solvability criteria, providing a constructive approach to solving such BVPs. The significance of this research lies in its contribution to the broader theory of BVPs in complex domains. The results obtained not only extend classical boundary conditions but also offer a systematic framework for dealing with higher-order equations. The interplay between different boundary conditions is explored in detail, revealing new insights into the structure of solutions and their dependence on boundary data. Beyond the theoretical implications, our findings have potential applications in mathematical physics, fluid dynamics, and engineering, where such problems frequently arise in modeling physical phenomena. Future research may further extend these results to more general domains and nonlinear equations, enriching the field of complex analysis and partial differential equations.

Scalar Field Source Teleparallel Robertson–Walker F(T) Gravity Solutions

This paper investigates the teleparallel Robertson–Walker (TRW) F(T) gravity solutions for a scalar field source. We use the TRW F(T) gravity field equations (FEs) for each k-parameter value case added by a scalar field to find new teleparallel F(T) solutions. For k=0, we find an easy-to-compute F(T) solution formula applicable for any scalar field source. Then, we obtain, for k=−1 and +1 situations, some new analytical F(T) solutions, only for specific n-parameter values and well-determined scalar field cases. We can find by those computations a large number of analytical teleparallel F(T) solutions independent of any scalar potential V(ϕ) expression. The V(ϕ) independence makes the FE solving and computations easier. The new solutions will be relevant for future cosmological applications in dark matter, dark energy (DE) quintessence, phantom energy and quintom models of physical processes.

Output tracking problem for a class of 3-D hyperbolic PDEs

The output tracking problem (OTP) refers to the procedure of determining an input control that ensures the system output follows a specified trajectory. In this paper, we investigate the OTP for controlled systems governed by a class of 3-D hyperbolic partial differential equations (PDEs), aiming for an exact alignment between the system output and the desired trajectory. Utilizing backstepping techniques, semigroup theory, the Hahn-Banach theorem, Laplace transforms, and Green's functions, our approach begins by establishing the OTP solution as a bounded feedback law, with Green's functions allowing the solution of a Cauchy-Euler equation. This is followed by deriving a detailed formulation for the complete state of the corresponding control problem. Finally, the approach provides an explicit formula for the OTP solution, using the desired trajectory and the solutions of specific kernel PDEs.

Monotonicity formula and Liouville-type theorem for stable solutions of weighted fourth order elliptic equation

Monotonicity formula and Liouville-type theorem for stable solutions of weighted fourth order elliptic equation

Whole space case for solution formula of Korteweg type fluid motion in R3

In this paper we consider the solution formula of linearized diffusive capillary model of Korteweg type without surface tension in three-dimensional Euclidean space ℝ3 using Fourier transform. Firstly, we construct the matrix of differential operators from the model problem. Then, we apply Fourier transform to the matrix. In the third step, we consider the resolvent problem of model problem. Finally, we find the solution formula of velocity and density by using inverse Fourier transform. For the further research we can consider not only estimating the solution operator families of the Korteweg theory of capillarity but also estimating the optimal decay for solution to the non-linear problem.

UJI LARUTAN SAFRANIN – PHOSPHATE BUFFERED SALINE SEBAGAI REAGEN ALTERNATIF HITUNG ERITROSIT DAN TROMBOIT

Manual examination of RBC and PLT was carried out using a microscope using different reagents, namely Hayem and Rees Ecker. We formulated a diluent solution consisting of PBS and Safranin for RBC and PLT examination at the same time. Therefore, we tried to test our Safranin – PBS solution formula so that it could be used to determine RBC and PLT simultaneously. The research involved 10 respondents with blood testing units. RBC and PLT were measured using a hematology analyzer (control) and Safranin – PBS pH 6.8 and 7.0 (treatment). The results of microscopic observations of PLT appear to be more clearly purple red compared to RBC which is pink. Reference method RBC 4.76 ± 0.62 x 106/µL. Safranin – PBS pH 6.8, namely 2.74 ± 0.82 x 106/µL (p-value = 0.000), Safranin – PBS pH 7.0, namely 3.05 ± 0.48 x 106/µL (p-value = 0.000). Regression equation Safranin – PBS pH 6.8 Y = -0.0425x + 348.69 (r2 = 0.0041), Safranin – PBS pH 7.0 Y = -0.242x + 402.8 (r2 = 0.1084) . It was concluded that the Safranin - PBS formula cannot be recommended because it still has different results and does not have the same method as the reference.

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.

A higher-order quadratic NLS equation on the half-line

The well-posedness of the initial-boundary value problem for higher-order quadratic nonlinear Schrödinger equations on the half-line is studied by utilizing the Fokas solution formula for the corresponding linear problem. Using this formula, linear estimates are derived in Bourgain spaces for initial data in spatial Sobolev spaces on the half-line and boundary data in temporal Sobolev spaces suggested by the time regularity of the linear initial value problem. Then, the needed bilinear estimates are derived and used for showing that the iteration map defined via the Fokas solution formula is a contraction in appropriate solution spaces. Finally, well-posedness is established for optimal Sobolev exponents in a way analogous to the case of the initial value problem on the whole line with solutions in classical Bourgain spaces.

Calculation of Green's Function for Poisson's Equation in Plane Polar Coordinates

A new calculation of Green's function for Poisson's equation in plane polar coordinates is presented. The method consists in first calculating the solution to the simpler problem, but with the same Green's function, that is obtained with the homogenization of the boundary conditions and then inferring Green's function by comparing this calculated solution with Green's solution formula. Depending on how the solution to the simplified problem is calculated, Green's function may result as an integral or an infinite series, but it is finally presented in a closed form, because it is possible to calculate the integral or the sum of this series.

Large Time Behavior for Solutions to the Anisotropic Navier–Stokes Equations in a 3D Half-space

Abstract We consider the large time behavior of the solution to the anisotropic Navier–Stokes equations in a $3$D half-space. Investigating the precise anisotropic nature of linearized solutions, we obtain the optimal decay estimates for the nonlinear global solutions in anisotropic Lebesgue norms. In particular, we reveal the enhanced dissipation mechanism for the third component of velocity field. We notice that, in contrast to the whole space case, some difficulties arises on the $L^{1}(\mathbb{R}^{3}_{+})$-estimates of the solution due to the nonlocal operators appearing in the linear solution formula. To overcome this, we introduce suitable Besov-type spaces and employ the Littlewood–Paley analysis on the tangential space.

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.

Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions

We address the Poincaré-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order n≥3, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also p-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.

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.