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In this paper we consider a system of linear bars, spherically jointed at their ends. For each bar one end is linked to the origin. We discuss the equations from which one obtains the deviation of the origin, and some possible optimizations concerning the minimum displacement of the origin and the minimum force in one bar, which are the main goals of the paper. The optimization is performed considering that for two bars one end is unknown; that is, the angles between the bars and the axes are unknown. It is proved that it is difficult to obtain an analytical solution in the general case, and the problem can be discussed only by numerical methods. A numerical case is also studied and some comments concerning the results are given. [ABSTRACT FROM AUTHOR]

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory. [ABSTRACT FROM AUTHOR]

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach. [ABSTRACT FROM AUTHOR]

With the frequent occurrence of financial risks, financial innovation supervision has become an important research issue, and excellent regulatory strategies are of great significance to maintain the stability and sustainable development of financial markets. Thus, this paper intends to analyze the financial regulation strategies through evolutionary game theory. In this paper, the delayed replication dynamic equation and the non-delayed replication dynamic equation are established, respectively, under different reward and punishment mechanisms, and their stability conditions and evolutionary stability strategies are investigated. The analysis finds that under the static mechanism, the internal equilibrium is unstable, and the delay does not affect the stability of the system, while in the dynamic mechanism, when the delay is less than a critical value, the two sides of the game have an evolutionary stable strategy, otherwise it is unstable, and Hopf bifurcation occurs at threshold. Finally, some numerical simulation examples are provided, and the numerical results show the correctness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The eigenvalues of complex q-Sturm–Liouville boundary value problems are the focus of this paper. The coefficients of the corresponding q-Sturm–Liouville equation provide the lower bounds on the real parts of all eigenvalues, and the real part of the eigenvalue and the coefficients of this q-Sturm–Liouville equation provide the bounds on the imaginary part of each eigenvalue. [ABSTRACT FROM AUTHOR]

The paper considers the a posteriori error estimates for fully discrete approximations of time-dependent Poisson–Nernst–Planck (PNP) equations, which provide tools that allow for optimizing the choice of each time step when working with adaptive meshes. The equations are discretized by the Backward Euler scheme in time and conforming finite elements in space. Overcoming the coupling of time and the space with a full discrete solution and dealing with nonlinearity by taking G-derivatives of the nonlinear system, the computable, robust, effective, and reliable space–time a posteriori error estimation is obtained. The adaptive algorithm constructed based on the estimates realizes the parallel adaptations of time steps and mesh refinements, which are verified by numerical experiments with the time singular point and adaptive mesh refinement with boundary layer effects. [ABSTRACT FROM AUTHOR]

This paper deals with statistical submanifolds and a family of statistical connections on them. The geometric structures such as the second fundamental form, curvatures tensor, mean curvature, statistical Ricci curvature and the relations among them on a statistical submanifold of a statistical manifold equipped with F -statistical connections are examined. The equations of Gauss and Codazzi of F -statistical connections are obtained. Such structures when the statistical submanifolds are conjugate symmetric are discussed. We present a inequality for statistical submanifolds in real space forms with respect to F -statistical connections. Also, we obtain a basic inequality involving statistical Ricci curvature and the squared F -mean curvature of a statistical submanifold of statistical manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized sine-Gordon equation ψ t x = (1 + a ∂ x 2) sin ψ and the sinh-Poisson equation u x x + u y y + σ sinh u = 0 , where a is a real parameter, and σ is a positive parameter. Under different conditions, e.g., a = 0 , a ≠ 0 , and σ > 0 , the periods of the periodic wave solutions for the above two equations are discussed. By the transformation of variables, the generalized sine-Gordon equation and sinh-Poisson equations are reduced to planar dynamical systems whose first integral includes trigonometric terms and exponential terms, respectively. We successfully handle the trigonometric terms and exponential terms in the study of the monotonicity of the period function of periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fourth-order three-point boundary value problem. Despite the fact that the corresponding Green's function changes its sign on the square of its definition, we obtain the existence of at least one positive and decreasing solution under some suitable conditions. The results are based on the classical Krasosel'skii's fixed point theorem in cones. Then, we impose some sufficient conditions that allow us to deduce nonexistence results. In the end, some examples are given in order to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on fractional stochastic differential equations (FSDEs) with a stochastic forcing term, i.e., to FSDE, we add a stochastic forcing term. Using the implicit scheme of Euler's approximation, the conditions for the existence and uniqueness of the solution of FSDEs with a stochastic forcing term are established. Such equations can be applied to considering FSDEs with a permeable wall. [ABSTRACT FROM AUTHOR]

In this paper, we make use of the simultaneous decomposition of eight quaternion matrices to study the solvability conditions and general solutions to a system of two-sided coupled Sylvester-type quaternion matrix equations A i X i C i + B i X i + 1 D i = Ω i , i = 1 , 2 , 3 , 4. We design an algorithm to compute the general solution to the system and give a numerical example. Additionally, we consider the application of the system in the encryption and decryption of color images. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2 , let (P n (k)) n be the k-generalized Pell sequence which starts with 0 , ... , 0 , 1 (k terms), and each term afterwards is given by P n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . In this paper, we determine all solutions of the Markov equation x 2 + y 2 + z 2 = 3 x y z , with x, y, and z being k-generalized Pell numbers. This paper continues and extends a previous work of Kafle, Srinivasan and Togbé, who found all Markov triples with Pell components. [ABSTRACT FROM AUTHOR]

We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the permissible range of exponents in a specific equation. Using L p − L q estimates for solutions to the corresponding linear fractional σ –evolution equations with vanishing right-hand sides, and applying a fixed-point argument, the existence of small data solutions is established for some admissible range of powers (p 1 , p 2 , ... , p k) . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local and semilocal convergence of an iterative method for solving nonlinear systems of equations. We first establish the conditions under which these methods converge locally to the solution. Then, we extend our analysis to examine the semilocal convergence of these methods, considering their behavior when starting from initial guesses that are not necessarily close to the solution. Iterative approaches for solving nonlinear systems of equations must take into account the radius of convergence, computable upper error bounds, and the uniqueness of solutions. These points have not been addressed in earlier studies. Moreover, we provide numerical examples to demonstrate the theoretical findings and compare the performance of these methods under different circumstances. Finally, we conclude that our examination offers a significant understanding of the convergence characteristics of previous iterative techniques for solving nonlinear equation systems. [ABSTRACT FROM AUTHOR]

In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not assumed that the real part of the symbol of the differential-difference operator contained in the equation has a constant sign. Previously, it was possible to remove that substantial restriction (i.e., the specified sign constancy) only for the case where the nonlocal term (i.e., the translated potential) is unique. In the present paper, we consider the case of the general-kind one-variable nonlocal potential, i.e., equations with an arbitrary amount of translated terms. No commensurability assumptions are imposed on the translation lengths. The following results are presented: We find a condition relating the coefficients at the nonlocal terms of the investigated equation and the length of the translations, providing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation. [ABSTRACT FROM AUTHOR]

Split quaternions have various applications in mathematics, computer graphics, robotics, physics, and so on. In this paper, two useful, real representations of a split quaternion matrix are proposed. Based on this, we derive their fundamental properties. Then, via the real representation method, we obtain the necessary and sufficient conditions for the existence of solutions to two split quaternion matrix equations. In addition, two experimental examples are provided to show their feasibility. [ABSTRACT FROM AUTHOR]

We deal with the following singular perturbation Kirchhoff equation: − ϵ 2 a + ϵ b ∫ R 3 | ∇ u | 2 d y Δ u + Q (y) u = | u | p − 1 u , u ∈ H 1 (R 3) , where constants a , b , ϵ > 0 and 1 < p < 5 . In this paper, we prove the uniqueness of the concentrated solutions under some suitable assumptions on asymptotic behaviors of Q (y) and its first derivatives by using a type of Pohozaev identity for a small enough ϵ. To some extent, our result exhibits a new phenomenon for a kind of Q (x) which allows for different orders in different directions. [ABSTRACT FROM AUTHOR]

The conventional approach of the local convergence analysis of an iterative method on R m , with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in R m spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects. [ABSTRACT FROM AUTHOR]

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach's applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using ( L 2 and L ∞ ) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods' unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors. [ABSTRACT FROM AUTHOR]

This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions and nonlocal derivatives, which are not specified, in contrast to the traditional approach to a point, but are the results of averaging over small but finite intervals of the independent variable are introduced. Classical equations of mathematical physics preserve the traditional form but include nonlocal functions. These equations are supplemented with additional equations that link nonlocal and traditional functions. The proposed approach results in continuous solutions of the classical singular problems of mathematical physics. The problems of a string and a circular membrane loaded with concentrated forces are used to demonstrate the procedure. Analytical results are supported with experimental data. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for a three-component generalization of Camassa–Holm equation (G3CH equation henceforth) with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell's equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell's equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell's equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze a specific fourth-order matrix spectral problem involving four potentials and two free nonzero parameters and construct an associated integrable hierarchy of bi-Hamiltonian equations within the zero curvature formulation. A hereditary recursion operator is explicitly computed, and the corresponding bi-Hamiltonian formulation is established by the so-called trace identity, showing the Liouville integrability of the obtained hierarchy. Two illustrative examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations with four components and two adjustable parameters. [ABSTRACT FROM AUTHOR]

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy (∥ ρ ∥ L ∞ (0 , T ; L ∞) + ∥ u ∥ C ([ 0 , T ] ; L 3) < ∞) , then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over [ 0 , T ] . Notably, our analysis accommodates the presence of an initial vacuum. [ABSTRACT FROM AUTHOR]

In this paper, we establish a structural inequality of the ∞-subLaplacian ▵ 0 , ∞ in a class of the semi-simple Lie group endowed with the horizontal vector fields X 1 , ... , X 2 n . When 1 < p ≤ 4 with n = 1 and 1 < p < 3 + 1 n − 1 with n ≥ 2 , we apply the structural inequality to obtain the local horizontal W 2 , 2 -regularity of weak solutions to p-Laplacian equation in the semi-simple Lie group. Compared to Euclidean spaces R 2 n with n ≥ 2 , the range of this p obtained is already optimal. [ABSTRACT FROM AUTHOR]

This paper explores solitary wave solutions arising in the deformations of a hyperelastic compressible plate. Explicit traveling wave solution expressions with various parameters for the hyperelastic compressible plate are obtained and visualized. To analyze the perturbed equation, we employ geometric singular perturbation theory, Melnikov methods, and invariant manifold theory. The solitary wave solutions of the hyperelastic compressible plate do not persist under small perturbations for wave speed c > − β k 2 . Further exploration of nonlinear models that accurately depict the persistence of solitary wave solution on the significant physical processes under the K-S perturbation is recommended. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate solvability of systems of constrained matrix equations in the form of constrained minimization problems. The main novelty of this paper is the unification of solutions of considered matrix equations with corresponding minimization problems. For a particular case we extend some well-known results and give several new results for the weak Drazin inverse. The main characterizations of the Drazin inverse, group inverse and Moore–Penrose inverse are obtained as consequences. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the setting and analysis of an optimal control problem for a two-phase system composed of two non-linearly coupled Chan–Hilliard-type equations. The model describes the evolution of a tumor cell fraction and a nutrient-rich extracellular water volume fraction. The main objective of this paper is the identification of the system's physical parameters, such as the viscosities and the proliferation rate, in addition to the controllability of the system's unknowns. For this purpose, we introduce an adequate cost function to be optimized by analyzing a linearized system, deriving the adjoint system, and defining the optimality condition. Eventually, we provide a numerical simulation example illustrating the theoretical results. Finally, numerical simulations of a tumor growing in two and three dimensions are carried out in order to illustrate the evolution of such a clinical situation and to possibly suggest different treatment strategies. [ABSTRACT FROM AUTHOR]

In this paper, a singular non-Newton polytropic filtration equation under the initial-boundary value condition is revisited. The finite time blow-up results were discussed when the initial energy E (u 0) was subcritical ( E (u 0) < d ), critical ( E (u 0) = d ), and supercritical ( E (u 0) > d ), with d being the potential depth by using the potential well method and some differential inequalities. The goal of this paper is to give a finite time blow-up result if E (u 0) is independent of d. Moreover, the explicit upper bound of the blow-up time is obtained by the classical Levine's concavity method, and the precise lower bound of the blow-up time is derived by applying an interpolation inequality. [ABSTRACT FROM AUTHOR]

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed. [ABSTRACT FROM AUTHOR]

( R , S )-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the ( R , S )-(skew) symmetric solutions to the quaternion matrix equation A X B = C . We use a real representation A τ to obtain the necessary and sufficient conditions for A X B = C to have ( R , S )-(skew) symmetric solutions and derive the solutions when it is consistent. We also derive the least-squares ( R , S )-(skew) symmetric solution to the above matrix equation. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the challenge of identifying an unknown source in a space-time fractional diffusion-wave equation. Through an analysis of the exact solution, it becomes evident that the problem is ill-posed. To address this, we employ both the Tikhonov regularization method and the Quasi-boundary regularization method, aiming to restore the stability of the solution. By adhering to both a priori and a posteriori regularization parameter choice rules, we derive error estimates that quantify the discrepancies between the regularization solutions and the exact solution. Finally, we present numerical examples to illustrate the effectiveness and stability of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, the authors propose the application of the Gibbs–Appell equations to obtain the equations of motion in the case of a mechanical system that has elements with a micro-polar structure, containing voids. Voids can appear as a result of the processing or manufacturing of the parts, or can be intentionally introduced. This research involves a model of the considered solid material containing voids. To determine the dynamic behavior of such a system, the Gibbs–Appell (GA) method is used to obtain the evolution equations, as an alternative to Lagrange's classical description. The proposed method can be applied to any mechanical system consisting of materials with a micro-polar structure and voids. The study of such systems is interesting because the literature shows that even a reduce number of small voids can produce significant variations in physical behavior. The proposed method requires a smaller number of mathematical operations. To apply this method, the acceleration energy is calculated, which is then used to derive the equations. The method comes with advantages in the application to multibody systems having the mentioned properties and, in particular, in the study of robots and manipulators. Using the GA method, it is necessary to do a fewer differentiation operations than applying the Lagrange's equations. This leads to a reduced amount of computation for obtaining the evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O (λ N − γ) in space and O (τ γ) in time, where γ ∈ (0 , 1 ] from the assumption ∥ A γ − 1 2 Q 1 2 ∥ L 2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents an efficient two-step hybrid block method (ETHBM) to obtain an approximate solution to the FitzHugh–Nagumo problem. The considered partial differential equation model problems are semi-discretized, reducing them to equivalent ordinary differential equations using the method of lines. In order to evaluate the effectiveness of the proposed ETHBM, three numerical examples are presented and compared with the results obtained through existing methods. The results demonstrate that the proposed ETHBM produces more efficient results than some other numerical approaches in the literature. [ABSTRACT FROM AUTHOR]

The equation G (x , x) = y ˜ , where G : X × X → Y , and X , Y are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones E + , M + of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a "model" equation, namely, g (t , t) = 0 , where a continuous map g : E + × E + → M + is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with "simpler" ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the "model" equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical linear algebra analytical study of some schemes for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation. Both 1D and 2D finite difference discretizations in space are proposed with semi-implicit and implicit discretizations on time. We prove that our proposed numerical solutions converge to continuous solutions. [ABSTRACT FROM AUTHOR]

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation's assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems. [ABSTRACT FROM AUTHOR]

The present paper investigates the following inhomogeneous generalized Hartree equation i u ˙ + Δ u = ± | u | p − 2 | x | b (I α ∗ | u | p | · | b) u , where the wave function is u : = u (t , x) : R × R N → C , with N ≥ 2 . In addition, the exponent b > 0 gives an unbounded inhomogeneous term | x | b and I α ≈ | · | − (N − α) denotes the Riesz-potential for certain 0 < α < N . In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term | x | b for certain b > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild  L p (D) -solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers  α ∈ (1 2 , 1 ]  by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in  L ∞ (0 , T ; L p (D))  space for all  p > 2 2 α − 1 , which is consistent with the deterministic case. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on an abstract iterative procedure for solving nonsmooth constrained generalized equations. This procedure employs both the property of weak point-based approximation and the approach of searching for a feasible inexact projection on the constrained set. Utilizing the contraction mapping principle, we establish higher order local convergence of the proposed method under the assumption of metric regularity property which ensures that the iterative procedure generates a sequence converging to a solution of the constrained generalized equation. Under strong metric regularity assumptions, we obtain that each sequence generated by this procedure converges to a solution. Furthermore, a restricted version of the proposed method is considered, for which we establish the desired convergence for each iterative sequence without a strong metric subregularity condition. The obtained results are new even for generalized equations without a constraint set. [ABSTRACT FROM AUTHOR]

Local convergence analysis is mostly carried out using the Taylor series expansion approach, which requires the utilization of high-order derivatives, not iterative methods. There are other limitations to this approach, such as the following: the analysis is limited to finite-dimensional Euclidean spaces; no a priori computable error bounds on the distance or uniqueness of the solution results are provided. The local convergence analysis in this paper positively addresses these concerns in the more general setting of a Banach space. The convergence conditions involve only the operators in the methods. The more important semi-local convergence analysis not studied before is developed by using majorizing sequences. Both types of convergence analyses are based on the concept of generalized continuity. Although we study a certain class of methods, the same approach applies to extend the applicability of other schemes along the same lines. [ABSTRACT FROM AUTHOR]

In this paper, the Kadometsev–Petviashvili equation and the Bargmann system are obtained from a second-order operator spectral problem L φ = (∂ 2 − v ∂ − λ u) φ = λ φ x . By means of the Euler–Lagrange equations, a suitable Jacobi–Ostrogradsky coordinate system is established. Using Cao's method and the associated Bargmann constraint, the Lax pairs of the differential equations are nonlinearized. Then, a new kind of finite-dimensional Hamilton system is generated. Moreover, involutive representations of the solutions of the Kadometsev–Petviashvili equation are derived. [ABSTRACT FROM AUTHOR]

This paper is dedicated to exploring the existence, uniqueness and Ulam stability analysis applied to a specific class of mathematical equations known as nonlinear impulsive Volterra Fredholm integro-dynamic adjoint equations within finite time scale intervals. The primary aim is to establish sufficient conditions that demonstrate Ulam stability for this particular class of equations on the considered time scales. The research methodology relies on the Banach contraction principle, Picard operator and extended integral inequality applicable to piecewise continuous functions on time scales. To illustrate the applicability of the findings, an example is provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations. [ABSTRACT FROM AUTHOR]

In this paper, motivated by a biodegradable stent problem, we consider a parabolic differential equation on graphs. This kind of equation describes the stent material degradation in time. Since stents are mesh-like structures with thin struts, they can be modelled using a simple structure called the one-dimensional curved rod model. In this way, we obtain a graph-like domain for our parabolic equation. Here, we prove the regularity estimate for the unique solution of the equation, together with corresponding estimates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative solution of a quadratic matrix equation, and the maximal positive definite solution of the equation X + A ∗ X − 1 A = Q . We study the recent iterative methods for computing the solution to the above specific type of equations and propose more effective modifications of these iterative methods. In addition, we make comments and comparisons of the existing methods and show the effectiveness of our methods by illustration examples. [ABSTRACT FROM AUTHOR]

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B 1 a Green function is constructed in the cases c > 0 , c ∉ − N , where c is the coefficient in front of u in the boundary condition ∂ u ∂ n + c u = f . To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λ u + c u = f . Elliptic boundary value problems for Δ m u = 0 in B 1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δ u = 0 , Δ 2 u = 0 and Δ 3 u = 0 in B 1 as well as some additional results from the theory of spherical functions are proposed. [ABSTRACT FROM AUTHOR]