Explicit Formula For Solution Research Articles

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.

Representation of solutions to fuzzy linear fractional differential equations with a piecewise constant argument

This study marks the first exploration of fuzzy linear fractional differential equations with a piecewise constant argument (FLFDEs-PCA), incorporating the concept of Caputo’s type gH-differentiability with the order α ∈ (0, 1]. Such problems are noteworthy as they represent hybrid systems, blending the characteristics of continuous and discrete dynamical systems and integrating aspects from both differential and difference equations. The primary objective of this research is to establish a standardized framework for deriving explicit solution formulas for FLFDEs-PCA under various scenarios. Additionally, illustrative examples are provided to demonstrate the practical implications of our theoretical findings.

Delayed analogue of three-parameter pseudo-Mittag-Leffler functions and their applications to Hilfer pseudo-fractional time retarded differential equations

Delayed analogue of three-parameter pseudo-Mittag-Leffler functions and their applications to Hilfer pseudo-fractional time retarded differential equations

Time optimal problems on Lie groups and applications to quantum control

<abstract><p>In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control (<sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $.</p> <p>In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by (<sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.</p></abstract>

Soliton solutions and a bi-Hamiltonian structure of the fifth-order nonlocal reverse-spacetime Sasa-Satsuma-type hierarchy via the Riemann-Hilbert approach

<p>Our objective is to explore the intricacies of a nonlinear nonlocal fifth-order scalar Sasa-Satsuma equation in reverse spacetime which is rooted in a nonlocal $ 5 \times 5 $ matrix AKNS spectral problem. Starting with this spectral problem, we derive both local and nonlocal symmetry relations through rotations within a defined group. We then formulate a specific type of Riemann-Hilbert problem, facilitating the generation of soliton solutions. These solutions are generated by utilizing vectors that reside in the kernel of the matrix Jost solutions. Under the condition where reflection coefficients are null, the jump matrix reduces to the identity, leading to soliton solutions via the corresponding Riemann-Hilbert problem. The explicit formulas of these soliton solutions enable a comprehensive exploration of their dynamics.</p>

The Riemann–Hilbert approach for the Chen–Lee–Liu equation with higher-order poles

This paper considers the Riemann–Hilbert problem (RHP) with higher-order poles for the Chen–Lee–Liu equation. By assuming the scattering coefficients have Nth order zero points, we obtain expressions of these scattering coefficients, the theta condition, and parallelization conditions between Jost solutions and their derivatives. Based on these parallelization conditions, it yields generalizations of residue conditions, which are essential for solving the RHP with higher-order poles. The explicit formulae for solutions of CLL equation are reconstructed from asymptotic behaviors of the RHP. As an example, we present the second-order pole soliton and illustrate that parameter b0 plays a principal role to determine the structures of solutions of the CLL equation.

On the Solution of the Dirichlet Problem for Second-Order Elliptic Systems in the Unit Disk

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method.

Time domain model order reduction of discrete-time bilinear systems by discrete Walsh functions

In this paper, the discrete Walsh functions are first used for time domain model order reduction (MOR) of discrete-time bilinear systems. Firstly, in terms of the discrete Walsh functions, the reduced order system is generated by means of the explicit solution formula for the discrete-time bilinear system. Meanwhile, we expand the discrete-time bilinear system under space spanned by the discrete Walsh functions. Then the reduced order system is produced by the orthogonal projection matrix that is defined by discrete Walsh coefficient matrix. Moreover, MOR method via discrete Walsh functions is investigated for discrete-time bilinear systems with nonzero initial conditions, where nonzero initial condition is also utilized to create the orthogonal projection matrix. Theoretical analyses show that the outputs of the resulting reduced order systems can match the first several outputs of the original system in finite intervals. Finally, an example is given to illustrate the effectiveness of the proposed methods.

Explicit Nth order solutions of Fokas–Lenells equation based on revised Riemann–Hilbert approach

We develop a revised Riemann–Hilbert problem (RHP) to the Fokas–Lenells (FL) equation with a zero boundary condition, satisfying the normalization condition, and the potential of the FL equation is recovered from the asymptotic behavior of RHP when the spectral parameter goes to zero. Under the reflection-less situation, we consider the RHP with 2N simple poles and two Nth order poles, respectively, and obtain the explicit formulas of Nth order soliton and positon solutions. As applications, the first-order soliton, the second-order soliton, and positon are displayed. Additionally, the collisions of N solitons are studied, and the phase shift and space shift are displayed.

Initial-boundary value problem for the Hirota equation posed on a finite interval

In this paper, we turn our attention to a nonhomogeneous initial boundary value problem for Hirota equation posed on a bounded interval (0,L). In particular, the explicit solution formula of linear nonhomogeneous boundary value problem is established by Laplace transform. Using space L2(0,T;H0s−1(0,1)), which is preparing for trilinear estimates and Lions-Magenes interpolation theorem, we prove the local existence, uniqueness, and Lipschitz continuous in C(0,T;Hs(0,1))∩L2(0,T;Hs+1(0,1)) corresponding to the initial and boundary data. Moreover, the local solution extends to a global one by a priori bound.

Discrete diffusion-type equation on regular graphs and its applications

ABSTRACT We derive an explicit formula for the fundamental solution to the discrete-time diffusion equation on the -regular tree in terms of the discrete I-Bessel function. We then use the formula to derive an explicit expression for the fundamental solution to the discrete-time diffusion equation on any -regular graph X. Going further, we develop three applications. The first one is to derive a general trace formula that relates the spectral data on X to its topological data. Though we emphasize the results in the case when X is finite, our method also applies when X has a countably infinite number of vertices. As a second application, we obtain a closed-form expression for the return time probability distribution of the uniform random walk on any -regular graph. The expression is obtained by relating to the uniform random walk on a -regular graph. We then show that if is a sequence of -regular graphs whose number of vertices goes to infinity and which satisfies a certain natural geometric condition, then the limit of the return time probability distributions from is equal to the return time probability distribution on the tree . As a third application, we derive formulas which express the number of distinct closed irreducible walks without tails on a finite graph X in terms of moments of the spectrum of its adjacency matrix.

Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems

<p style='text-indent:20px;'>In this paper, we focus on learning optimal parameters for PDE-based image denoising and decomposition models. First, we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to machine learning approaches. The numerical experiments correlate with our theoretical model settings and show a reduction of computing time in contrast to the Rudin-Osher-Fatemi model. Second, we introduce a new regularized image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.</p>

Численный метод решения задачи Коши для одного дифференциального уравнения с дробной производной Капуто

The Cauchy problem for differential equations with fractional derivatives is used in many spheres of science and technology. It was the reason for the development of various methods for its solution, both analytic and approximate ones. The search of an exact solution of differential equations with fractional derivatives by analytic methods is a complex and ineffective task; for this reason, an attempt to solve the considered problem approximately is undertaken in this paper. gated on the segment [0, T]. The method of finite differences which is relatively primary to implement is used for the numerical solution. A difference scheme approximating the Cauchy problem with the first order is constructed on a uniform grid. The difference problem is studied for stability and convergence with a fixed value of the function α(t). It is shown that the numerical solution of the problem converges to the exact solution in the first order. Explicit recurrent formulas for the numerical solution are obtained. A computational experiment upon analysis of the numerical solution of the Cauchy problem is carried out. It is shown on the basis of the computational experiment that if we take the average value for α(t), the first order exactness takes place.

The logistic equation in the context of Stieltjes differential and integral equations

In this paper, we introduce logistic equations with Stieltjes derivatives and provide explicit solution formulas. As an application, we present a population model which involves intraspecific competition, periods of hibernation, as well as seasonal reproductive cycles. We also deal with various forms of Stieltjes integral equations, and find the corresponding logistic equations. We show that our work extends earlier results for dynamic equations on time scales, which served as an inspiration for this paper.

Toward uniform existence and convergence theorems for three-scale systems of hyperbolic PDEs with general initial data

Uniform existence of solutions to initial-value problems and convergence of appropriately filtered solutions are proven for a special class of three-scale singular limit equations, without any restriction on the initial data. The uniform existence is proven using a novel system of energy estimates. The convergence result is based on a detailed analysis of the fastest-scale oscillations, which unlike in two-scale systems have no explicit solution formula.

Duality for Stieltjes differential and integral equations

We propose new definitions of adjoint equations to first-order linear Stieltjes differential and integral equations. We investigate the existence and uniqueness of their solutions, provide explicit solution formulas, and obtain corresponding versions of Lagrange's identity. We show that our results are compatible with known results for dynamic equations on time scales, which are included as a special case.

Representation and finite time stability of solution and relative controllability of conformable type oscillating systems

In this paper, we firstly introduce the concept of conformable matrix sine and cosine functions, which help us to seek explicit formula of solutions to a conformable type linear oscillating system by using the variation of constants method. Secondly, we establish some sufficient conditions to guarantee the finite time stability results and give sufficient and necessary conditions to examine that a linear conformable system is relatively controllable. Further, we apply fixed point theorem to prove relatively controllable result for a semilinear conformable system. Finally, examples are presented to illustrate the validity of the main theorems.

The Robin and Neumann problems for the nonlinear Schrödinger equation on the half-plane

This work studies the initial-boundary value problem (ibvp) of the two-dimensional nonlinear Schrödinger equation on the half-plane with initial data in Sobolev spaces and Neumann or Robin boundary data in appropriate Bourgain spaces. It establishes well-posedness in the sense of Hadamard by using the explicit solution formula for the forced linear ibvp obtained via Fokas’s unified transform, and a contraction mapping argument.

A direct method for the simultaneous updating of finite element mass, damping and stiffness matrices

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices and with , for , , and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that . Problem OAP: Given real symmetric matrices and , find such that , where is the solution set of Problem IQEP and is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data.