Dimensional Partial Differential Equations Research Articles

Mixed convection flow of radiated Sutterby nanofluid by stretchable cylinder with irreversibility and heat generation

Entropy generation has gained consideration of researchers due to its applications. Applications of entropy occur in chillers, desert collars, refrigerators, and all type of heat transfer devices. Due to vast range of applications of entropy, in this study irreversibility in Sutterby nanofluid flow by stretchable cylinder is discussed. While reporting momentum equation magnetic field and mixed convection effects are considered. Influences of radiation, Joule heating, and heat source are deliberated in the expression for thermal energy. Chemical reaction impact is taken in the modeling of concentration equation. Irreversibility for the considered flow is obtained by utilizing thermodynamics second law. Dimensional partial differential equations (PDEs) representing the flow are transformed to dimensionless ordinary differential equations(ODEs) through similarity transformations. Solution of ODEs is obtained via NDSolve code of Mathematica. Impact of flow parameters on velocity, entropy, temperature, Bejan number, and concentration are studied graphically. Engineering quantities are analyzed numerically. It is noticed through achieved results that velocity drops for higher magnetic variable while upsurges for higher curvature variable. Thermal field boosts for higher magnetic and heat source parameter. For rising diffusion and radiation variables entropy improves.

Invariant analysis, exact solutions, and conservation laws of time fractional thin liquid film equations

This present paper investigates Lie symmetry analysis, one-dimensional optimal system, exact solutions and conservation laws of the (2 + 1)-dimensional time fractional thin liquid film equations (TFTLFE) with Riemann–Liouville fractional derivative. Explicitly, we obtain six vector fields and the one-dimensional optimal system admitted by TFTLFE. Then, we perform the symmetry reductions with the help of Erdélyi–Kober fractional differential operator and (2 + 1)-dimensional TFTLFE is reduced into (1 + 1)-dimensional fractional partial differential equations (FPDEs). Additionally, by means of compound variable transformation and the power series expansion method, the solution of reduced FPDEs is obtained and its convergence is verified. Moreover, we derive other solutions for the reduced equations taking advantage of the invariant subspace method. Furthermore, the conservation laws are also established utilizing generalized Noether's theorem. Finally, we construct the exact solution using the method of conservation laws.

Non-similar analysis of bioconvective stagnation-point flow toward a stretching surface in the presence of buoyancy force, viscous dissipation, and thermal radiation

Aim of this analysis is to investigate, bioconvective stagnation flow of an incompressible fluid across an expanding perpendicular surface in the existence of buoyancy force, viscous dissipation, and thermal radiation. The physical model is described in terms of nonlinear coupled partial differential equations (PDEs). The dimensional PDEs are transformed into dimensionless PDEs by using appropriate non-similar transformations. Local nonlinearity method (LNS) is used to generate high accuracy truncated ordinary differential equations (ODEs) that are used to approximate dimensionless PDEs. Using well-known techniques like finite-difference-based bvp4c, the ODEs are numerically formulated. This research also investigates how several physical, nondimensional parameters affect the profiles of velocity, temperature, and concentration, for both assisted and opposed flow cases. The comparison and range tables describe the validity and range of the presented results, respectively. The tabulated results indicate that enhancement in radiation parameter increases the local Nusselt number, Sherwood number, and the surface drag force.

Heat and Mass Transfer of Boundary Layer Jeffrey Ternary Hybrid Nanofluid Flow Over Porous Wedge With Surface‐Catalysed Chemical Reactions: An ANFIS Model

The current study employs machine learning (ML) techniques to investigate the heat and mass transport of Jeffrey ternary hybrid nanofluid (THNF). Various influential factors, including magnetic field, thermal radiation, viscous dissipation, activation energy, thermophoresis, Brownian motion and surface‐catalysed chemical reactions, are considered in the analysis. The porous moving wedge supports and influences the flow pattern. For thermal enhancement, three different nanoparticles, namely, Ag (silver), CuO (copper oxide) and SWCNT (single − walled carbon nanotubes), diluted in the regular fluid (blood). Initially, the mathematically modelled partial differential equations (PDEs) are solved numerically by the help of the shooting method. The suitable nondimensional similarity transformations are implied to convert the dimensional PDEs to nondimensional ordinary differential equations (ODEs). The reduced dimensionless nonlinear coupled ODEs are solved using ode‐45 in MATLAB. The impact of φAg, φCuO, φSWCNT in Cfx, Nux, Shx, is displayed in cone plots. It is found that, on increasing φAg, φCuO, φSWCNT, the heat transfer and mass transfer rate gets enhanced in THNF than the hybrid nanofluid (HNF) and nanofluid (NF). The influence of nondimensional parameters over f″(0), θ′(0), ϕ′(0), G′(0) and H′(0) is displayed in 3D contour plots for THNF and HNF. The radiation parameter (R), Brownian motion parameter (Nb), thermophoresis parameter (Nt) and volume fraction parameters φAg, φCuO, φSWCNT boost the energy transfer rate. Finally, the obtained numerical results are split into training and validation datasets that are used in designing the adaptive neuro‐fuzzy inference system (ANFIS) ML model. Five different ANFIS models are developed with the help of nondimensional parameters affecting f″(0), θ′(0), ϕ′(0), G′(0) and H′(0) to predict the physical quantities of dragging force (Cfx), energy transfer rate (Nux), the rate of mass transport (Shx) and mass fluxes for chemical species and , respectively. The training and checking errors attained the convergence less than 2 × 10−4 and 0.2, respectively, for all the five factors. The present ANFIS models have good balance between fitting the training data and generalising to new data, which can make accurate predictions irrespective of variations. The numerical and predicted ANFIS f″(0), θ′(0), ϕ′(0), G′(0) and H′(0) for training data, checking data and results are demonstrated in regression plots. From regression plots, we can observe that the Pearson correlation coefficient attains the positive correlation with R value closer to one. Hence, the developed ANFIS model shows the minimal error in predictions with high degree of accuracy of forecast in THNF flow.

Well-posedness and finite element approximation of mixed dimensional partial differential equations

In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.

Lie symmetry analysis, numerical solutions by spectral method and travelling wave solutions of a (5+1)-dimensional partial differential equation

ABSTRACT The purpose of this research is to investigate the Lie symmetries and the numerical solution of the (5 + 1)-dimensional evolutionary differential equation using the Chebyshev spectral method. Also, the exact solution of this equation is presented by the traveling wave method. These equations play a very important role in mathematical physics and engineering sciences. The implemented algorithm is quite efficient and practically suitable for these problems. The use of Maple and MATLAB software allows us to perform complex and boring calculations. The most distinctive aspect of this method is that no prescribed assumptions are required, and enormous computational effort is reduced and rounding errors are avoided. Our proposed methods for checking and solving the considered equation show that this method is very accurate and practical.

A deep-genetic algorithm (deep-GA) approach for high-dimensional nonlinear parabolic partial differential equations

We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their corresponding backward stochastic differential equations (BSDEs). Recognizing the sensitivity of the solver to the initial guess selection, we embed a genetic algorithm (GA) into the solver to optimize the selection. We aim to achieve faster convergence for the nonlinear PDEs on a broader interval than deep-BSDE. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes (BS) equation with default risk and the Hamilton-Jacobi-Bellman (HJB) equation. We compare the results of our method with those of the deep-BSDE and show that our method provides comparable accuracy with significantly improved computational efficiency.

Numerical solutions of an option pricing rainfall weather derivatives model

Weather derivatives are financial products used to cover non catastrophic weather events with a weather index as the underlying asset. In this work, we derive a rainfall weather derivative price model, based in the assumption that the rainfall dynamics follows a Ornstein-Uhlenbeck process. To calculate the price of the option we arrive at a two dimensional time dependent partial differential equation, where the spatial variables are the rainfall index and the total rainfall. Appropriate boundary conditions are suggested and they differ from the boundaries presented in literature in similar contexts. To compute the approximate solutions of the partial differential equation, we propose an explicit numerical method in order to deal efficiently with the different choices of the coefficients involved in the equation, that depend on the rainfall defice (or excess) and on the precipitation (amount of rain). Being an explicit numerical method, it will be conditionally stable and we discuss the stability region of the numerical method and its order of convergence. In the end we examine two test cases where the parameters of the model presented are estimated based on precipitation data.

A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations

A computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.

Unsteady magneto bioconvective Sutterby nanofluid flow: Influence of g-Jitter effect

This study is driven by the potential applications of a microgravity environment in various sectors and fields of engineering, especially in the manufacturing of semiconductors. An electrically conducting bioconvective Sutterby nanofluid flow was considered around an exponentially stretching surface through g-Jitter and nonlinear thermal radiation effects. The dimensional partial differential equations (PDEs) that govern the fluid flow system are changed into nondimensional PDEs through nonsimilar transformations. Once these PDEs have been linearized using Quasilinearization, the implicit finite difference scheme is used to solve them numerically. A MATLAB code was constructed for the numerical calculations and for analysing the results via graphs. These findings indicate that the greater values of amplitude ε1 result in a steep rise in the velocity and surface drag near the sheet's wall for assisting buoyancy. An enhancement of approximately 108 % in the drag coefficient is observed when ε1(the amplitude in g-Jitter) jumps to 1 from 0. The percentile increments in heat transport strength due to the rise of radiation Rd from 1 to 2 is 80 %, and due to the intensification of the temperature difference ratio φw from 1.5 to 2 is about 167 %. Elevating Peclet number Pe led to the depletion of the microbial boundary layer thickness. Oxytactic microbes' presence impacts the enhanced mass transport strength of nanoparticles. The current findings have been validated by comparing them to previously published results, and they have been shown to be in perfect agreement with those results.

Finite element analysis of MHD naturally convective flow past an exponentially accelerated plate with viscous dissipation

The MHD finite element analysis naturally convective flow across an exponentially accelerating plate with viscous dissipation is investigated numerically. In this process dimensional partial differential equations are changed to dimensional less form. The Galerkin finite element method is emerged to solve non-dimensional partial differential equations. Obtained results are effectively depicted through the utilization of graphs, allowing for a comprehensive understanding of the impact that different physical parameters on primary velocity, transverse velocity, temperature, and concentration profiles. The study effectively differentiated between the present and prior findings, ultimately demonstrating a strong consensus of excellent agreement between the results. The key findings of this study are: The primary velocity increases on growing the values of Thermal Grashof number, Solutal Grashof number, Eckert number, and Dufour number and decreases on climb up the values of Magnetic parameter, Prandtl number, Schmidt number, and Hall parameter. The secondary velocity increases on raising the values of Thermal Grashof number and Solutal Grashof number and diminishes on enhancing the value of Hall parameter. The temperature increases on increasing the values Eckert number and Dufour number and decreases on improving the value of Prandtl number. The concentration decreases on increasing the value of Schmidt number.

Solutions of Time Fractional (1 + 3)-Dimensional Partial Differential Equations by the Natural Transform Decomposition Method (NTDM)

The current study employs the natural transform decomposition method (NTDM) to test fractional-order partial differential equations (FPDEs). The present technique is a mixture of the natural transform method and the Adomian decomposition method. For the purpose of checking the precis of our technique, some examples are offered, and the series solutions of these equations are introduced by using NTDM. The outcome shows that the suggested approach is very active and straightforward for obtaining a series solutions of FPDEs and is more accurate if we compare it with existing methods.

Periodic, n-soliton and variable separation solutions for an extended (3+1)-dimensional KP-Boussinesq equation

An extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation is studied in this paper to construct periodic solution, n-soliton solution and folded localized excitation. Firstly, with the help of the Hirota’s bilinear method and ansatz, some periodic solutions have been derived. Secondly, taking Burgers equation as an auxiliary function, we have obtained n-soliton solution and n-shock wave. Lastly, we present a new variable separation method for (3+1)-dimensional and higher dimensional models, and use it to derive localized excitation solutions. To be specific, we have constructed various novel structures and discussed the interaction dynamics of folded solitary waves. Compared with the other methods, the variable separation solutions obtained in this paper not only directly give the analytical form of the solution u instead of its potential u_y, but also provide us a straightforward approach to construct localized excitation for higher order dimensional nonlinear partial differential equation.

Du Fort-Frankel Finite Difference Scheme for Solving of Oxygen Diffusion Problem inside One Cell

In this paper, we use the well-known Du Fort-Frankel finite difference scheme to solve the oxygen diffusion problem inside one cell which is modeled by an initial moving boundary value problem for one dimensional time-dependent partial differential equation. The main problem consists in tracking the moving boundary that represents the oxygen penetration depth inside the cell. We explore the possibilities of numerical approximation of the problem posed by the different formulations. Some numerical experiments are also provided with comparisons with analytical solution. The theoretical analysis is given for the numerical scheme. It is shown that all the results obtained by this method are compared with earlier authors.

Nonlinear radiation effects on water-based nanofluid containing CNTs subject to heat source/ sink past a wedge

In this paper, a two-dimensional and incompressible laminar flow comprised of water-based carbon nanotubes over convectively heated moving wedge under the magnetic field and nonlinear radiation and heat production/ absorption is investigated. The base nanofluid (water) contains single wall carbon nanotubes (SWCNTs) and multiple walls carbon nanotubes (MWCNTs). In order to convert the dimensional nonlinear partial differential equations in nondimensional nonlinear ordinary differential form, an adequate set of similarity variables had been used. These set of equations and boundary conditions are evaluated by the implementation of RKF-45 (Runge–Kutta–Fehlberg fourth-fifth) order scheme. The influence of several physical parameters on particular nanoparticle’s volume friction, temperature and velocity ratio parameter, heat source/ sink parameter, nonlinear radiative constraint, exponent constant, magnetic factor, Eckert and Biot numbers is studied. An opposite behavior of volume fraction and velocity ratio parameters on velocity and energy profiles is achieved.

An application of soliton solutions of a differential equation in the progression of aortic dissection

Aortic dissection is a serious pathology involving the vessel wall of the aorta with significant societal impact. To understand aortic dissection, we explain the role of the dynamic pathology in the absence or presence of structural and/or functional abnormalities. We frame a differential equation to evaluate the impact of mean blood pressure on the aortic wall and prove the existence and uniqueness of its solution for homeostatic recoil and relaxation for infinitesimal aortic tissue. We model and analyze a generalized (3 + 1)‐dimensional nonlinear partial differential equation for aortic wave dynamics. We use the Lie group of transformations on this nonlinear evolution equation to obtain invariant solutions, traveling wave solutions including solitons. We find that abnormalities in the dynamic pathology of aortic dissection act as triggers for the progression of disease in the early stage through the formation of soliton‐like pulses and their interaction. We address the role of unstable wavefields in waveform dynamics when waves are unidirectional. Moreover, the notion of dynamic pathology within the domain of vascular geometry may explain the evolution of aneurysms in cerebral arteries and cardiomyopathies even in the absence of anatomical and physiological abnormalities.

Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations

Abstract In this study, a new structure for the septic B-spline collocation algorithm in n-dimensional is presented as a continuation of generating B-spline functions in n-dimensional to solve mathematical models in n-dimensional. The septic B-spline collocation algorithm is displayed in three forms: one dimensional, two dimensional, and three dimensional. In various domains, these constructs are essential for solving mathematical models. The effectiveness and correctness of the suggested method are demonstrated using a few two- and three-dimensional test problems. The proposed new structure provides better results than other methods because it deals with a larger number of points than the field. To create comparisons, we use different numerical approaches accessible in the literature.

Approximate symmetry of time-fractional partial differential equations with a small parameter

In the sense of Riemann–Liouville fractional derivative, an approximate symmetry method of (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed. By first inserting the power series expansion of dependent variable in the small parameter into the fractional PDEs and then separating the equation with respect to the small parameter to get a coupled system of fractional PDEs, we define the pth-order approximate symmetries by the Lie symmetries of the first p+1 coupled fractional PDEs. After performing symmetry reductions for the coupled system of fractional PDEs, we find that zero-order reduced equation coincides with the fractional ordinary differential equation corresponding to the fractional PDE without containing the small parameter while the solutions of higher-order reduced equation can be obtained by solving linear variable coefficient fractional ordinary differential equations. We first exemplify the whole procedure of computing approximate symmetry by studying a time-fractional KdV equation with a small parameter. Specifically, we obtain the higher-order approximate symmetries and perform approximate symmetry reductions by means of an optimal system of one-dimensional approximate Lie subalgebras. Moreover, we construct an explicit power series solution and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Then in order to make the proposed approximate symmetry method be further convincing, we study an anomalous diffusion equation with a small parameter and construct an analytical solution in terms of the Mittag-Leffler function.

Symmetry reduction and exact solutions of the (3+1)-dimensional nKdV-nCBS equation

In this paper, the exact solutions of the (3+1)-dimensional nKdV-nCBS equation are investigated by various approaches. Firstly, according to Lie group theory, the (3+1)-dimensional nKdV-nCBS equation is reduced to two (1+1)-dimensional partial differential equations. Secondly, three different types of exact solutions are obtained by the homoclinic test approach, including the singular kinked-type periodic solitary wave solutions, the kinked solitary wave solutions and the kinked-cross periodic wave solutions. The diagrams of the solutions for the specified parameters are given. Furthermore, taking one reduced equation as an example, the N-soliton solutions are derived. And under some constraints, higher-order soliton solution can degenerate into lower-order soliton solution, which is a specific degeneration phenomenon of this equation.

An efficient quantum partial differential equation solver with chebyshev points

Differential equations are the foundation of mathematical models representing the universe’s physics. Hence, it is significant to solve partial and ordinary differential equations, such as Navier–Stokes, heat transfer, convection–diffusion, and wave equations, to model, calculate and simulate the underlying complex physical processes. However, it is challenging to solve coupled nonlinear high dimensional partial differential equations in classical computers because of the vast amount of required resources and time. Quantum computation is one of the most promising methods that enable simulations of more complex problems. One solver developed for quantum computers is the quantum partial differential equation (PDE) solver, which uses the quantum amplitude estimation algorithm (QAEA). This paper proposes an efficient implementation of the QAEA by utilizing Chebyshev points for numerical integration to design robust quantum PDE solvers. A generic ordinary differential equation, a heat equation, and a convection–diffusion equation are solved. The solutions are compared with the available data to demonstrate the effectiveness of the proposed approach. We show that the proposed implementation provides a two-order accuracy increase with a significant reduction in solution time.