A Semi-analytical Study on Non-Linear Partial Differential Equations in Different Enzyme Kinetics with Amperometric Biosensors

Differential invariants of systems of two nonlinear elliptic partial differential equations by Lie symmetry method

Abstract It has long been recognized that the theory of nonlinear elasticity provides a rich framework for a large variety of issues of interest to applied mathematicians. In particular, researchers with primary interest in nonlinear partial differential equations have been attracted to this area of continuum mechanics. However, the detailed theoretical background giving rise to the governing partial differential equations is not always familiar to non-specialists. The purpose of the present expository note is to attempt to alleviate this situation by describing a variety of nonlinear partial differential equations that have been found to govern the deformations of anti-plane shear and plane strain for isotropic incompressible hyperelastic solids in equilibrium.

This paper focuses on the fourth-order nonlinear Boussinesq equation (FONBE), which describes the interaction mechanisms of solutions in shallow-water waves. The FONBE has broad applications across various fields of physics and engineering, including heat transfer, fluid convection, and modeling of underwater volcanic activity, ocean currents, seafloor movements, and other hydrothermal processes. The Kumar-Malik method (KMM) and the modified exponential function method (MEFM) are employed to derive different types of soliton solutions. Analytical forms expressed through Jacobi elliptic, hyperbolic, trigonometric, exponential, and rational functions are obtained. Graphical representations further support the analytical findings, providing a clear understanding of the solution behaviors. These results highlight the effectiveness of the proposed methodology in addressing nonlinear challenges in mathematics and engineering, offering improvements over previous studies. The applied techniques are robust, efficient, and adaptable to a wide class of nonlinear partial differential equations. The novelty of this work lies in the generation of several new soliton solutions through the application of analytical approaches that have not been previously applied to this equation.

The FitzHugh–Nagumo (FHN) equation in one dimension is solved in this paper using an improved physics-informed neural network (PINN) approach. Examining test problems with known analytical solutions and the explicit finite difference method (EFDM) allowed for the demonstration of the PINN’s effectiveness. Our study presents an improved PINN formulation tailored to the FitzHugh–Nagumo reaction–diffusion system. The proposed framework is efficiently designed, validated, and systematically optimized, demonstrating that a careful balance among model complexity, collocation density, and training strategy enables high accuracy within limited computational time. Despite the very strong agreement that both methods provide, we have demonstrated that the PINN results exhibit a closer agreement with the analytical solutions for Test Problem 1, whereas the EFDM yielded more accurate results for Test Problem 2. This study is crucial for evaluating the PINN’s performance in solving the FHN equation and its application to nonlinear processes like pulse propagation in optical fibers, drug delivery, neural behavior, geophysical fluid dynamics, and long-wave propagation in oceans, highlighting the potential of PINNs for complex systems. Numerical models for this class of nonlinear partial differential equations (PDEs) may be developed by existing and future model creators of a wide range of various nonlinear physical processes in the physical and engineering sectors using the concepts of the solution methods employed in this study.

We study an element agglomeration coarsening strategy that requires data redistribution at coarse levels when the number of coarse elements becomes smaller than the number of MPI processes used on the finest level. The overall procedure generates coarse elements (general unstructured unions of fine grid elements) within the framework of element-based algebraic multigrid methods (or AMGe) studied previously. The AMGe-generated coarse spaces have the ability to exhibit approximation properties of the same order as the fine-level spaces since by construction they contain the piecewise polynomials of the same order as on the fine level. These approximation properties are key for the successful use of AMGe in multilevel solvers for nonlinear partial differential equations as well as for multilevel Monte Carlo (MLMC) simulations. The ability to coarsen without being constrained by the number of MPI processes, as described in the present paper, allows to improve the scalability of these solvers as well as the overall MLMC method. The paper illustrates this latter fact with detailed scalability study of MLMC simulations applied to model Darcy equations with a stochastic log-normal permeability field.

ABSTRACT In this paper, the stochastic Davey–Stewartson mathematical model of hydrodynamics, nonlinear optics, and plasma physics is considered. This model is impressive in that it can describe complex multidimensional wave processes under the action of random factors, which is typical for natural physical systems. The primary aim of this study is to acquire and examine exact stochastic solutions of the Davey–Stewartson equation via an analytical method. The problem is initially decomposed into real and imaginary parts, yielding a system of nonlinear partial differential equations (NLPDEs). The system is then reduced to a set of linear equations and associated polynomial versions. The resulting linear system gives some sets of solutions with both the model parameters and the form of the proposed solution. An appropriate set of solutions is determined, and a wave transformation is performed to allow the solutions to be obtained. The Jacobi elliptic function expansion method, a powerful analytical method, is used to get exact solutions of the Davey–Stewartson equation. This method offers a wide range of solution forms, such as singular, periodic, and trigonometric waveforms. In addition, numerical solutions are established for the study of the influence of noise on the reached solutions, and the results are presented in terms of 3D, 2D, and contour plots based on parameters obtained by an analytical procedure. The results provide new exact solutions in a stochastic environment, highlighting the importance of the process used. These findings represent novel results never previously presented in the literature.

This paper investigates a physics-informed surrogate modeling framework for multi-phase flow in porous media based on the Fourier Neural Operator. Traditional numerical simulators, though accurate, suffer from severe computational bottlenecks due to fine-grid discretizations and the iterative solution of highly nonlinear partial differential equations. By parameterizing the kernel integral directly in Fourier space, the operator provides a discretization-invariant mapping between function spaces, enabling efficient spectral convolutions. We introduce a Dual-Branch Adaptive Fourier Neural Operator with a shared Fourier encoder and two decoders: a saturation branch that uses an inverse Fourier transform followed by a multilayer perceptron and a pressure branch that uses a convolutional decoder. Temporal information is injected via Time2Vec embeddings and a causal temporal transformer, conditioning each forward pass on step index and time step to maintain consistent dynamics across horizons. Physics-informed losses couple data fidelity with residuals from mass conservation and Darcy pressure, enforcing the governing constraints in Fourier space; truncated spectral kernels promote generalization across meshes without retraining. On SPE10-style heterogeneities, the model shifts the infinity-norm error mass into the 10−2 to 10−1 band during early transients and sustains lower errors during pseudo-steady state. In zero-shot three-dimensional coarse-to-fine upscaling from 30×110×5 to 60×220×5, it attains R2=0.90, RMSE = 4.4×10−2, and MAE = 3.2×10−2, with more than 90% of voxels below five percent absolute error across five unseen layers, while the end-to-end pipeline runs about three times faster than a full-order fine-grid solve and preserves water-flood fronts and channel connectivity. Benchmarking against established baselines indicates a scalable, high-fidelity alternative for high-resolution multi-phase flow simulation in porous media.

In this research work, we are interested in the discrete study of the blow-up time of the solution of certain nonlinear parabolic Partial Differential Equations (PDEs) subject to nonlinear boundary conditions. We have been able to establish the necessary and sufficient conditions under which the discrete solution of the problem blows up in a finite discrete time and, at the same time, we have given an estimate of this blow up time. Also, using a convergence study, we showed that the discrete time and solution converge respectively to the continuous time and solution when the discretization steps in space and time tend towards zero. Finally, we illustrated our analysis with graphical representations and some numerical results.

Abstract This study explores the modified Benjamin–Bona–Mahony equation using the new extended direct algebraic approach, a powerful analytical technique for solving nonlinear partial differential equations. The proposed methodology yields a diverse spectrum of exact solutions, categorized into 12 distinct classes, including rational, hyperbolic, and trigonometric functions, as well as mixed periodic, singular, shock-singular, complex solitary-shock, and plane-wave solutions. These solutions are systematically derived and validated using Mathematica , demonstrating the reliability and effectiveness of the method. A comparative analysis with existing techniques underscores the consistency and superiority of the proposed approach. Additionally, the Hamiltonian function is constructed to examine the system’s conservation properties, ensuring the physical relevance of the obtained solutions. A comprehensive sensitivity analysis is performed to assess the model response to variations in parameters and initial conditions. To further illustrate the dynamical characteristics of the solutions, three-dimensional, two-dimensional, and contour plots are presented, offering deeper insights into their physical behavior. The results contribute to the larger study of nonlinear wave phenomena in engineering and applied sciences, providing a robust analytical framework for future research in soliton theory and mathematical physics.

Abstract This study examines mass and heat transfer in a permeable ternary nanofluid flow over a stretching sheet, considering the combined effects of a chemical reaction, Joule heating, an exponentially space-dependent heat source, and an inclined magnetic field. Three types of water-based nanofluids are analysed: mono (Cu), hybrid (Cu + Al 2 O 3 ), and ternary (Cu + Al 2 O 3 + Ag). The governing nonlinear partial differential equations are reduced using similarity transformations and solved numerically via MATLAB’s BVP4c method. The results reveal that ternary nanofluids exhibit superior thermal performance, with significantly higher temperature profiles compared to mono and hybrid nanofluids. The influence of key parameters is also investigated. Increased suction and velocity slip reduce thermal and concentration boundary layers, while higher Biot numbers and heat source intensity enhance temperature profiles. Additionally, Joule heating and magnetic field inclination intensify the heat transfer rate. These findings provide valuable insights for optimizing thermal systems in applications such as solar energy collectors, thermoelectric devices, and chemical processing industries.

ABSTRACT In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third‐order nonlinear partial differential equations with the type . We prove that there are two subclasses of equations admitting a local isometric immersion into the three‐dimensional Euclidean space for which the coefficients of the second fundamental form depend on a jet of finite order of , and furthermore, these coefficients are universal, namely, they are functions of and , independent of . Finally, we show that the generalized Camassa–Holm equation describing pseudospherical surfaces has a universal second fundamental form.

An exact analysis of Magnetohydrodynamics (MHD) free convection flow of electrically conducting and chemically reactive fluid moving through a vertical plate with ramped wall conditions is presented. An inclined uniform magnetic field was applied to the vertical plate to influence the flow behavior. The cross-diffusion effect, like thermophoresis (Soret), has also been considered. The Laplace transform method was used to solve the governed set of nonlinear partial differential equations in precise closed form. The study simulated the influence of various parameters, such as the heat sink, magnetic field, first-order chemical reaction parameter, thermal radiation parameter, permeability parameter, and applied magnetic force inclination parameter, on the temperature, concentration, velocity profiles, as well as Nusselt number, Sherwood number, and skin friction in terms of graphs and tables. The outcome showed that a rise in heat sink parameter increases the concentration profile, skin friction, and Nusselt number but decreases the velocity and temperature profiles as well as the Sherwood number. The velocity profile and skin friction both decrease as the magnetic field parameter is increased. The first-order chemical reaction is found to decrease the concentration of the fluid particle, while the thermal radiation parameter raises the temperature profile. The results obtained are in excellent agreement with those reported in previously published works.

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-momentum relation to determine new families of nonlinear partial differential relations. The Planck–de Broglie duality principle arises from Planck’s energy expression e=hν, de Broglie’s equation for momentum p=h/λ, and Einstein’s special relativity energy, where h is the Planck constant, ν and λ are the frequency and wavelength, respectively, of an associated wave having a wave speed w=νλ. The author has extended these relations to a family that is characterised by a second fundamental constant h′ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. In this note, we apply generalized Schrödinger operator relations and the power-law relations to generate a new family of nonlinear partial differential equations that are characterised by the constant κ=h′/h such that κ=0 corresponds to the Klein–Gordon equation. The resulting partial differential equation is unusual in the sense that it admits a stretching symmetry giving rise to both similarity solutions and simple harmonic travelling waves. Three simple solutions of the partial differential equation are examined including a separable solution, a travelling wave solution, and a similarity solution. A special case of the similarity solution admits zeroth-order Bessel functions as solutions while generally, it reduces to solving a nonlinear first-order ordinary differential equation.

This article presents the three well-known derivative operators—Caputo, Caputo-Fabrizio, and Atangana-Baleanu—to describe the solutions of the non-linear Fractional Partial Differential Equation (FPDE). A method called the Adomian Decomposition Method (ADM) is used to find series solutions in a semi-analytical way, using different transforms like Laplace, Elzaki, Sumudu, Aboodh, Mohand, Yang, Natural, and Shehu. The solutions obtained by the proposed method have precision and a high rate of convergence. We then verify the derived solutions numerically and graphically for both fractional and integer orders. Furthermore, the solutions under these transformations are the same. The proposed simulations show that as the number of iterations increases, the corresponding absolute error reduces. Moreover, fractional order solutions are converging to integer order solutions.

This paper presents a novel application of the improved modified extended tanh-function method to derive exact traveling wave solutions for the space-time fractional modified third- order Korteweg-de Vries (KdV) equation. The fractional derivatives are considered in the con- formable fractional derivative form. The proposed method systematically reduces the nonlinear fractional partial differential equation to an ordinary differential equation, which is then solved using a generalized ansatz. A specific kink-type solitary wave solution is obtained in closed form. The results demonstrate the robustness and efficiency of the method for handling com- plex nonlinear fractional differential equations, providing valuable new solutions that enhance our understanding of wave propagation in fractional media.

While recent advances have successfully integrated neural networks with physical models to derive numerical solutions, there remains a compelling need to obtain exact analytical solutions. The ability to extract closed-form expressions from these models would provide deeper theoretical insights and enhanced predictive capabilities, complementing existing computational techniques. In this paper, we study the nonlinear Gardner equation and the (2+1)-dimensional Zabolotskaya–Khokhlov model, both of which are fundamental nonlinear wave equations with broad applications in various physical contexts. The proposed models have applications in fluid dynamics, describing shallow water waves, internal waves in stratified fluids, and the propagation of nonlinear acoustic beams. This study integrates a modified generalized Riccati equation mapping approach and a novel generalized G′G-expansion method with neural networks for obtaining exact solutions for the suggested nonlinear models. Researchers are currently investigating potential applications of these neural networks to enhance our understanding of complex physical processes and to develop new analytical techniques. The proposed strategies incorporate the solutions of the Riccati problem into neural networks. Neural networks are multi-layer computing approaches including activation and weight functions among neurons in input, hidden, and output layers. Here, the solutions of the Riccati equation are allocated to each neuron in the first hidden layer; thus, new trial functions are established. We evaluate the suggested models, which lead to the construction of exact solutions in different forms, such as kink, dark, bright, singular, and combined solitons, as well as hyperbolic and periodic solutions, in order to verify the mathematical framework of the applied methods. The dynamic properties of certain wave-related solutions have been shown using various three-dimensional, two-dimensional, and contour visualizations. This paper introduces a novel framework for addressing nonlinear partial differential equations, with significant potential applications in various scientific and engineering domains.

Abstract This paper presents a robust computational technique to tackle the intricate nonlinear partial differential equations (PDEs) encountered in mathematical physics. The method is applied to the time-fractional Burgers-Huxley equation, where the time derivative is considered in the Liouville-Caputo sense. This equation, which combines the well-known Burgers and Huxley equations, describes the interplay of reaction, convection, and diffusion in transport phenomena and finds application in acoustics, turbulence theory, traffic flow, and hydrodynamics. The proposed method transforms this complex non-linear fractional PDE into a simple algebraic system. Its ability to handle the non-linear terms without perturbation, discretization, or the calculation of extraneous terms is a major advantage over available analytical approaches. Five different cases of the equation with diverse initial and boundary conditions are discussed. To demonstrate the accuracy and reliability of the semi-analytic approach, the obtained outcomes are compared with existing exact and analytical solutions in the literature, showing a strong level of agreement. Error analysis and the convergence criterion are also discussed.

Abstract. The flow of fluids within porous rocks is an important process with numerous applications in Earth sciences. Modeling the compaction-driven fluid flow requires the solution of coupled nonlinear partial differential equations that account for the fluid flow and the solid deformation within the porous medium. Despite the nonlinear relation of porosity and permeability that is commonly encountered, natural data show evidence of channelized fluid flow in rocks that have an overall layered structure. Layers of different rock types have discontinuous hydraulic and mechanical properties. We present numerical results obtained by a novel space-time method, which can handle discontinuous initial porosity (and permeability) distributions efficiently. The space-time method enables straightforward coupling to models of mass transport for trace elements. Our results indicate that, under certain conditions, the discontinuity of the initial porosity influences the distribution of incompatible trace elements, leading to sharp concentration gradients and large degrees of elemental enrichment. Finally, our results indicate that the enrichment of trace elements depends not only on the channelization of the flow but also on the interaction of fluid-filled channels with layers of different porosity and permeability.

ABSTRACT The third fractional 3D Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is examined in this paper, along with new waveforms and various analyses. This is important for understanding how waves move in plasma physics, shallow water, and nonlinear optics. We use a Galilean transformation to obtain the research output of this model. The planner dynamic system of the equation is also constructed, and all possible phase portrait analyses are described, including bifurcation and chaos. We observed chaotic, periodic, and quasi‐periodic behaviors by introducing a perturbed term for various parameter values. This study talks about multistability analysis, sensitivity analysis, and exact traveling wave solutions of the governing model. Fractal dimension, strange attractor, recurrence plot, power spectrum, return map, and Lyapunov exponent (LE) are some of the graphs that show how the model works. Additionally, this research work employs the unified solver technique to yield diverse solitary‐wave outcomes. We visually display the derived outcomes in 2D and 3D plots. We can conclude that these findings provide a solid foundation for further investigation and are valuable, useful, and reliable for dealing with future complex nonlinear problems. The approach employed in this work demonstrates a high level of reliability, robustness, and efficiency, making it suitable for addressing a vast area of nonlinear partial differential equations (NLPDEs) that have not been studied in any other research.