Search Results (61)

This paper develops several new generalizations of Gronwall–Bellman–Bihari-type integral inequalities. We establish three novel integral inequalities that extend classical results to more complex settings, including integrals with mixed linear and nonlinear terms, delayed (retarded) arguments, and general integral kernels. In the preliminaries, we review known Gronwall–Bellman–Bihari inequalities and useful lemmas. In the main results, we present at least three new theorems. The first theorem provides an explicit bound for solutions of an integral inequality involving a separable kernel function and a nonlinear (Bihari-type) term, significantly extending the classical Bihari inequality. The second theorem addresses integral inequalities with delayed arguments, showing that the delay does not enlarge the growth bound compared to the non-delay case. The third theorem handles inequalities with combined linear and nonlinear terms; using a monotone iterative technique, we prove the existence of a maximal solution that bounds any solution of the inequality. Rigorous proofs are given for all main results. In the Applications section, we illustrate how these inequalities can be applied to deduce qualitative properties of differential equations. As an example, we prove a uniqueness result for an initial value problem with a non-Lipschitz nonlinear term using our new inequalities. The paper concludes with a summary of results and a brief discussion of potential further generalizations. Our results provide powerful tools for researchers to obtain a priori bounds and uniqueness criteria for various differential, integral, and functional equations. It is important to note that the integral inequalities established in this work provide bounds on the solution under the assumption of its existence on the considered interval $[t_0,T]$. For nonlinear differential or integral equations where the nonlinearity $F$ fails to be Lipschitz continuous, solutions may develop movable singularities (blow-up) in finite time. The bounds derived from our Gronwall–Bellman–Bihari-type inequalities are valid only on the maximal interval of existence of the solution. Determining the region where solutions are guaranteed to be free of such singularities is a separate and profound problem, often requiring additional techniques such as the construction of Lyapunov functions or the use of differential comparison principles. The primary contribution of this paper is to provide sharp estimates and uniqueness criteria within the domain where a solution is known to exist a priori. Full article

This paper studies a mixed PDE containing the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables, which is encountered in geophysical fluid dynamics. The Lie group symmetry analysis of this highly nonlinear PDE is performed for the first time. An invariant point transformation is found that depends on fourteen arbitrary constants and preserves the form of the equation under consideration. One-dimensional symmetry reductions leading to self-similar and some other invariant solutions that described by single ODEs are considered. Using the methods of generalized and functional separation of variables, as well as the principle of structural analogy of solutions, a large number of new non-invariant closed-form solutions are obtained. In general, the extensive list of all exact solutions found includes more than thirty solutions that are expressed in terms of elementary functions. Most of the obtained solutions contain a number of arbitrary constants, and several solutions additionally include two arbitrary functions. Two-dimensional reductions are considered that reduce the original PDE in three independent variables to a single simpler PDE in two independent variables (including linear wave equations, the Laplace equation, the Tricomi equation, and the Guderley equation) or to a system of such PDEs. A number of specific examples demonstrate that the type of the mixed, highly nonlinear PDE under consideration, depending on the choice of its specific solutions, can be either hyperbolic or elliptic. To analyze the equation and construct exact solutions and reductions, in addition to Cartesian coordinates, polar, generalized polar, and special Lorentz coordinates are also used. In conclusion, possible promising directions for further research of the highly nonlinear PDE under consideration and related PDEs are formulated. It should be noted that the described symmetries, transformations, reductions, and solutions can be utilized to determine the error and estimate the limits of applicability of numerical and approximate analytical methods for solving complex problems of mathematical physics with highly nonlinear PDEs. Full article

Against the backdrop of Shenzhen’s high-density urban environment, the multifunctional design of water purification plants offers dual benefits: providing residents with urban green spaces while simultaneously mitigating NIMBY sentiments due to their inherent characteristics. Unlike traditional urban development, Shenzhen’s water purification plants integrate into residents’ daily lives. Therefore, optimizing the built environment and road network structure to enhance residents’ perceptions of proximity benefits while reducing NIMBY (Not In My Backyard effect) sentiments holds significant implications for the city’s sustainable development. To address this question, this study adopted the following three-step mixed-methods approach: (1) It examined the relationships among residents’ YIMBY (Neighboring Benefits Effect) and NIMBY perceptions, perceptions of park spaces atop water purification plants, and perceptions of accessibility through questionnaire surveys and structural equation modeling (SEM), establishing a scoring framework for comprehensive YIMBY and NIMBY perceptions. (2) Random forest models and Shapley Additive Explanations (SHAP) analysis revealed nonlinear relationships between the built environment and composite YIMBY and NIMBY perceptions. (3) Spatial syntax analysis categorized the upgraded road network around the water purification plant into grid-type, radial-type, and fragmented-type structures. Scatter plot fitting methods uncovered relationships between these road network types and resident perceptions. Finally, negative perceptions were mitigated by optimizing path enclosure and reducing visual obstructions around the water purification plant. Enhancing neighborhood benefits—through improved path safety and comfort, increased green spaces and resting areas, optimized path networks, and diversified travel options—optimized the built environment. This approach proposes design strategies to minimize NIMBY perceptions and maximize YIMBY perceptions. Full article

Shock–accelerated interfaces between fluids of different densities are prone to Richtmyer–Meshkov-type instabilities, whose evolution is strongly influenced by the incident shock Mach number. In this study, we present a systematic numerical investigation of the Mach number effect on the instability growth at a light–heavy fluid layer. The governing dynamics are modeled using the compressible multi-species Euler equations, and the simulations are performed with a high-order modal discontinuous Galerkin method. This approach provides accurate resolution of sharp interfaces, shock waves, and small-scale vortical structures. A series of two-dimensional simulations is carried out for a range of shock Mach numbers impinging on a sinusoidally perturbed light–heavy fluid interface. The results highlight the distinct stages of instability evolution, from shock–interface interaction and baroclinic vorticity deposition to nonlinear roll-up and interface deformation. Quantitative diagnostics—including circulation, enstrophy, vorticity extrema, and mixing width—are employed to characterize the instability dynamics and to isolate the role of Mach number in enhancing or suppressing growth. Particular attention is given to the mechanisms of vorticity generation through baroclinic torque and compressibility effects. Moreover, the analysis of controlling parameters, including Atwood number, layer thickness, and initial perturbation amplitude, broadens the parametric understanding of shock-driven instabilities. The results reveal that increasing shock Mach number markedly enhances vorticity generation and accelerates interface growth, while the resulting nonlinear morphology remains strongly sensitive to variations in Atwood number and perturbation amplitude. Full article

We present an analytical and numerical study of nonlinear wave localization in a one-dimensional rhombic (diamond) waveguide array that combines forward- and backward-propagating channels. This mixed-index configuration, realizable through Bragg-type couplers or corrugated waveguides, produces a tunable spectral gap and supports nonlinear self-localized states in both transmission and forbidden-band regimes. Starting from the full set of coupled-mode equations, we derive the effective evolution model, identify the role of coupling asymmetry and nonlinear coefficients, and obtain explicit soliton solutions using the method of multiple scales. The resulting envelopes satisfy a nonlinear Schrödinger equation with an effective nonlinear parameter $\theta$, which determines the conditions for soliton existence ($\theta >0$) for various combinations of focusing and defocusing nonlinearities. We distinguish solitons formed outside and inside the bandgap and analyze their dependence on the dispersion curvature and nonlinear response. Direct numerical simulations confirm the analytical predictions and reveal robust propagation and interactions of counter-propagating soliton modes. Order-of-magnitude estimates show that the predicted effects are accessible in realistic integrated photonic platforms. These results provide a unified theoretical framework for soliton formation in mixed-index lattices and suggest feasible routes for realizing controllable nonlinear localization in Bragg-type photonic structures. Full article

Nonlinear mixed integro-differential equations (NM-IDEs) of the third kind present a complex challenge during solving initial value problems (IVPs), particularly after converting them from standard forms. In this work, we address the existence and uniqueness of a type of NM-IDEs employing Picard’s method. Additionally, we estimate the solution using the homotopy analysis method (HAM) and analyze the convergence of the approach. To demonstrate the credibility of the theoretical results, various applications are given, and the numerical results are displayed in a group of figures and tables to highlight that solving IVPs by first converting them to NM-IDEs and using the HAM is a computationally efficient approach. Full article

In this work, we propose a method for the shape optimization of frame structures using a mixed analytical–numerical approach. The goal is to achieve a uniform-strength frame structure, ensuring optimal material utilization and weight minimization. The optimization is performed in two calculation steps. The first step uses an analytical model based on the Timoshenko beam theory, where appropriate mathematical steps give a uniform-strength shape of the entire structure. Depending on the type of cross-section analyzed, the exact uniform-strength profile of each element is derived by solving for three parameters related to the forces and moments acting on the element. These parameters are obtained by solving a nonlinear system of equations, which includes the external and internal kinematic constraints of the structure, as well as equilibrium equations for each element. However, the solution obtained using the one-dimensional theory is limited in areas affected by boundary effects, such as the interconnection regions between elements and those near the supports, for a decay distance at least equal to the characteristic diameter of the section. To address this limitation, the second optimization step involves incorporating solutions that account for a triaxial stress field. This is typically carried out by discretizing the structure using the finite element method. The frame geometry obtained from the previous analytical solution is constructed, and the regions affected by boundary effects are optimized using the Biological Growth Method (BGM). This is an iterative, bio-inspired method modeled on the growth of trees, which increases trunk diameter in proportion to the loads experienced. The method is applied simultaneously to all regions where three-dimensional effects are significant, with the aim of achieving uniform strength in areas influenced by boundary effects. An important aspect of applying the BGM is maintaining the topology of the initial mesh, which is ensured through the use of mesh morphing techniques. The results of the two-step optimization process are shown on simple geometries involving few elements, and on more complex geometries of mechanical interest. Full article

Using the deep result of Ran & Reunrings, we generalize existing results for tripled fixed points. In contrast to the previously known results for tripled fixed points of maps with or without the mixed monotone property in partially ordered complete metric spaces, we demonstrate that it is possible to obtain results for the existence and uniqueness of such points for arbitrary maps with a type of monotonicity, with the partial ordering in the Cartesian product arising from the maps itself. We prove theorems that ensure the existence and uniqueness for tripled fixed points for maps with different types of monotone properties. We obtain sufficient conditions for the existence and uniqueness of systems of three nonlinear matrix equations. The obtained results are illustrated by solving systems of matrix equations. Full article

In this paper, we use an existing fixed point iterative scheme to approximate a class of generalized $\alpha$-nonexpansive mapping in Banach spaces. We also prove weak and strong convergence results for the mapping using the AG iterative scheme. An example of a generalized $\alpha$-nonexpansive mapping is given to show the validity of the claims. We apply the main results to the approximation of solution of a mixed type Voltera–Fredholm functional nonlinear integral equation and to the spread of HIV modeled in terms of a fractional differential equation of the Caputo type. Full article

For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions. Full article

This paper investigates the stability of the $F^{\ast }$ iterative algorithm applied to strongly pseudocontractive mappings within the context of uniformly convex Banach spaces. The study leverages both analytic and numerical methods to demonstrate the convergence and stability of the algorithm. In comparison to previous works, where weak-contraction mappings were utilized, the strongly pseudocontractive mappings used in this study preserve the convergence property, exhibit greater stability, and have broader applicability in optimization and fixed point theory. Additionally, this work shows that the type of mapping employed converges faster than those in earlier studies. The results are applied to a mixed-type Volterra–Fredholm nonlinear integral equation, and numerical examples are provided to validate the theoretical findings. Key contributions of this work include the following: (i) the use of strongly pseudocontractive mappings, which offer a more stable and efficient convergence rate compared to weak-contraction mappings; (ii) the application of the $F^{\ast }$ algorithm to a wider range of problems; and (iii) the proposal of future directions for improving convergence rates and exploring the algorithm’s behavior in Hilbert and reflexive Banach spaces. Full article

This paper investigates second-order functional dynamic equations with mixed nonlinearities on an arbitrary unbounded above-time scale, T. We will use a unified time scale approach and the well-known Riccati technique to derive oscillation criteria of the Nehari-type for second-order dynamic equations. The findings demonstrate a significant improvement in the literature on dynamic equations. The symmetry is beneficial and influential in defining the right style of study for the qualitative behavior of solutions to dynamic equations. We include an example to demonstrate the significance of our results. Full article

Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in $L_2\left[\Omega \right]\times C\left[0,T\right],T<1,$$\Omega$ is the position domain of integration and T is the time. The convergence error and the stability error are considered. Then, after using the separation technique, the MIE transforms into a system of Hammerstein integral equations (SHIEs) with time-varying coefficients. The nonlinear algebraic system (NAS) is obtained after using the degenerate method. New and special cases are derived from this work. Moreover, numerical results are computed using MATLAB R2023a software. Full article

This research investigates the application of supercritical carbon dioxide (CO₂) within carbon capture, utilization, and storage (CCUS) technologies to enhance oil-well production efficiency and facilitate carbon storage, thereby promoting a low-carbon circular economy. We simulate the flow of supercritical CO₂ mixed with associated gas (flow rates 3–13 × 10⁴ Nm³/d) in a miniature venturi tube under high temperature and high-pressure conditions (30–50 MPa, 120–150 °C). Accurate fluid property calculations, essential for simulation fidelity, were performed using the R. Span and W. Wagner and GERG-2008 equations. A dual-parameter prediction model was developed based on the simulation data. However, actual measurements only provide fluid types and measurement data, such as pressure, temperature, and venturi differential pressure, to determine the liquid mass fraction (LMF) and total mass flow rate (m), presenting challenges due to complex nonlinear relationships. Traditional formula-fitting methods proved inadequate for these conditions. Consequently, we employed a Levenberg–Marquardt (LM) based neural network algorithm to address this issue. The LM optimizer excels in handling complex nonlinear problems with faster convergence, making it suitable for our small dataset. Through this approach, we formulated dual-parameter model equations to elucidate fluid flow factors, analyzing the impact of multiple parameters on the LMF and the discharge coefficient (C). The resulting model predicted dual parameters with a relative error for LMF of ±1% (Pc = 95.5%) and for m of ±1% (Pc = 95.5%), demonstrating high accuracy. This study highlights the potential of neural networks to predict the behavior of complex fluids with high supercritical CO₂ content, offering a novel solution where traditional methods fail. Full article

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications. Full article