Showing papers in "International Journal of Applied and Computational Mathematics in 2019"

Jacobi Elliptic Function Expansion Method for Solving KdV Equation with Conformable Derivative and Dual-Power Law Nonlinearity

Abstract: In this work, the KdV equation with conformable derivative and dual-power law nonlinearity is considered. It is exceedingly used as a model to depict the feeble nonlinear long waves in different fields of sciences. Furthermore, it explains the comparable effects of weak dispersion and weak nonlinearity on the evolvement of the nonlinear waves. Using the Jacobi elliptic function expansion method, new exact solutions of that equation have been found. As results, some obtained solutions behave as periodic traveling waves, bright soliton, and dark soliton.

A Three Species Food Chain Model with Fear Induced Trophic Cascade

Abstract: In ecology, predator–prey interaction is one of the most important factors. The effects of predators on prey population can be direct and deadly, or it may be indirect and non-consumptive. Recent experimental findings have explored that fear of predator (indirect effect) alone can change prey’s behavior including reproduction and foraging. Suraci et al. (Nature Communications, 7, 10698, 2016) experimentally showed that fear of large carnivore reduces mesocarnivore foraging, which benefits the mesocarnivore’s prey. They also showed that fear of large carnivore mediates a cascading effect in lower trophic level. In the present study, our aim is to observe how the cascading effects of fear in a tri-trophic food chain model influence the dynamics of the model. We propose a three-species food chain model incorporating the cost of fear into the predation rate of middle predator. We consider the fact that due to fear of the top predator, middle predator forage less. As a result, the predation rate of middle predator decreases which reduces the predation pressure on basal prey. Mathematical properties such as boundedness, persistence, equilibria analysis, local and global stability analysis of the model are investigated. We perform bifurcation analysis around interior equilibrium point of the system. We notice that cost of the fear in middle predator can stabilize an otherwise chaotic system. We also investigate the robustness of the stabilizing role of the fear parameter. We observe that system initiating from the different dynamical regime, fear ultimately drives the system towards stability. It is also found that for increasing the level of fear, the system enters into a stable state through multiple switching of dynamics. Our results suggest that cost of the fear in middle predator can stabilize the system and enhances persistence of the system. We illustrate our analytical results numerically. Finally our results qualitatively reflect the experimental findings of Suraci et al.

Numerical Investigation of the Time Fractional Mobile-Immobile Advection-Dispersion Model Arising from Solute Transport in Porous Media

Abstract: Evolution equations containing fractional derivatives can offer efficient mathematical models for determination of anomalous diffusion and transport dynamics in multi-faceted systems that cannot be precisely modeled by using normal integer order equations. In recent times, researches have found out that lots of physical processes illustrate fractional order characteristics that alters with time or space. The continuum of order in the fractional calculus permits the order of the fractional operator be accounted for as a variable. In the current research work, radial basis functions (RBFs) approximation is utilized for solving fractional mobile-immobile advection-dispersion (TF-MIM-AD) model in a bounded domain which is applied for explaining solute transport in both porous and fractured media. In this approach, firstly, the discretization process of the aforesaid equation with of convergence order $$\mathcal {O}(\delta t^{})$$ in the t-direction is described via the finite difference scheme for $$ 0< \alpha <1$$ . Afterwards, by help of the meshless methods based on RBFs, we will illustrate how to obtain the approximated solution. The stability and convergence of time-discretized scheme are also theoretically discussed in detail throughout the paper. Finally, two numerical instances are included to clarify effectiveness and accuracy of our proposed concepts which is investigated in the current research work.

A Numerical Approach for Multi-variable Orders Differential Equations Using Jacobi Polynomials

Abstract: In this work, we consider multi-variable orders differential equations (MVODE) and apply Jacobi polynomials such as the shifted Legendre and shifted Chebyshev polynomials for solving MVODE. Further we compare the given result.

Generalized Fibonacci Operational Collocation Approach for Fractional Initial Value Problems

Abstract: A numerical algorithm for solving multi-term fractional differential equations (FDEs) is established herein. We established and validated an operational matrix of fractional derivatives of the generalized Fibonacci polynomials (GFPs). The proposed numerical algorithm is mainly built on applying the collocation method to reduce the FDEs with its initial conditions into a system of algebraic equations in the unknown expansion coefficients. Output of the numerical results asserted that our developed algorithm is applicable, efficient and accurate.

Impact of Thermal Radiation on an Unsteady Casson Nanofluid Flow Over a Stretching Surface

Abstract: This article employs the analytical approach to examine the heat and mass transfer characteristics on an unsteady flow of Casson nanofluid past an elongated surface with thermal radiation effect. A perturbation technique is adopted to solve the governing equations of the flow. Copper, Silver and Ferrous nanoparticles are suspended in water based nanofluid. Impacts of physical parameters like nanoparticle volume fraction, thermal radiation, magnetic field, stretching parameter, heat source/sink, a chemical reaction on velocity, thermal and concentration attributes along with wall friction, heat, and mass transfer rates are demonstrated with the aid of graphs and tables. Dual nature is witnessed for Newtonian and non-Newtonian fluid cases. Obtained results demonstrate the volumetric size, shape and conductive property of the nanoparticle play an important role in enriching the effectiveness of convection heat transfer of nanofluids. Also, Casson nanofluid has a tendency to reduce the velocity of the fluid due to its higher viscidness.

A Two-Level Trade-Credit Approach to an Integrated Price-Sensitive Inventory Model with Shortages

Abstract: This paper deals with an economic order quantity model with deterioration for two different demand functions under two-level of trade-credit policy. The demand functions of the proposed model are two types: (i) exponential function of the price (ii) price with the negative power of constant. Shortages are allowed and fully backlogged as deterioration arises. A nonlinear constraint optimization problem is then formulated considering cost and profit parameters. The main objective of this paper is to find out the optimal selling price, optimal deteriorating length and optimal cycle length for the optimal total profit of the chain. Some theoretical as well as numerical outcomes are studied to show the validity of the proposed model. A sensitivity analysis is carried out to study the effect of changes of key parameters of the inventory system.

Nonlinear Radiation in Bioconvective Casson Nanofluid Flow

Abstract: We investigate the impact of nonlinear thermal radiation and variable transport properties on the two-dimensional flow of an electrically conducting Casson nanofluid containing gyrotactic microorganisms along a moving wedge. In some previous studies, it has been assumed that the fluid viscosity and thermal conductivity are temperature dependent. However, this study assumes that the fluid viscosity, thermal conductivity, and the nanofluid properties, are dependent on the solute concentration. Some experimental studies have shown that the viscosity and thermal conductivity of nanofluids are strongly dependent on the volume fraction of nanoparticles rather than just the temperature. The spectral local linearization method is used to solve the conservation equations. We compare our results with those in the literature, and we discuss the convergence and accuracy of the spectral local linearization method. The impact of some parameters on the skin friction, heat and microorganisms mass transport is discussed.

Numerical Simulation of Non-Newtonian Power-Law Fluid Flow in a Lid-Driven Skewed Cavity

Abstract: Non-Newtonian laminar fluid flow in a lid-driven skewed cavity has been studied numerically using power-law viscosity model. The governing two-dimensional unsteady incompressible Navier–Stokes equations were initially non-dimensionalized using appropriate transformation, and then the dimensionless form is transformed to generalized curvilinear coordinates to simulate complex geometry. The transformed equations are discretized using finite volume method with the collocated grid arrangement. The code is first validated against the existing benchmark results for two-dimensional lid-driven square cavity problem considering both Newtonian and non-Newtonian fluids. The validation has also been carried out for a lid-driven skewed cavity in the case of a Newtonian fluid. Then the code is applied to the skewed cavity problem involving non-Newtonian fluid flow which can be described by the power-law viscosity model. Moreover, grid independence test has been performed for a skewed cavity for different values of power-law index. In the present case, the skewness of the geometry has been changed by changing the skew angle for both shear-thinning and shear-thickening fluids. The consequent numerical results are presented in terms of the velocity as well as streamlines for the different values of the power-law index $$n=0.5$$ , 1 and 1.5, Reynolds number $$Re = 100, 200, 300$$ and 500 as well as for the different angles of the skewed cavity ( $$\alpha =15^{\circ }$$ to $$165^{\circ }$$ ).

Numerical Different Methods for Solving the Nonlinear Biochemical Reaction Model

Abstract: In this paper, we are interested in Sumudu Decomposition Method (SDM) and Hermite collocation method. SDM is considered as a mixture of Adomian decomposition method and the Sumudu transform method. Firstly, SDM and Hermite collocation method are applied to obtain the solutions of a nonlinear biochemical reaction model. In addition, our numerical results are compared opposite to those of the traditional numerical method that were obtained for these two methods. Signal flow graph and Simulink of MATLAB of this model are represented. The stability of equilibrium point is studied. Finally, we show that SDM and Hermite collocation method are extremely symmetry and similar.

Unsteady MHD Flow in a Porous Channel with Thermal Radiation and Heat Source/Sink

Abstract: The current study investigates the unsteady flow of a viscous, incompressible, electrically and thermally conducting fluid between two infinite parallel porous walls placed at y = 0 and y = a. It is assumed that the electrically conducting fluid is driven by a mutual action of the imposed pressure gradient, thermal buoyancy and heat source or sink. The fluid injection occurs at the left boundary wall of the flow channel whereas the fluid suction occurs at the right boundary wall of the flow channel. The flow occurs only when the fluid starts to move with time. The flow is subjective to a convective heat exchange with the surrounding boundaries. The unsteady system of the non-dimensional form of PDEs with the corresponding boundary conditions are solved by employing the explicit Finite Difference Scheme. In the presence of pertinent parameters, a precise movement of the electrically conducting fluid within the flow channel is shown graphically in the form of profiles such as velocity, temperature, skin friction coefficient and Nusselt number. Distinct from the other studies, in which the boundary layer system of PDEs are usually transformed into a system of ordinary differential equations by means of the similarity transformations, the current study provides an efficient numerical procedure to solve the system of PDEs without using the similarity transformations which illustrate the precise movement of the electrically conducting fluid within the flow channel.

A Modified Homotopy Perturbation Method for Nonlinear Singular Lane–Emden Equations Arising in Various Physical Models

Abstract: A modified homotopy perturbation method for solving a class of nonlinear Lane–Emden equations with boundary conditions arising in various physical models is proposed. The proposed algorithm is based on the homotopy perturbation method and integral form of the Lane–Emden equation. The integral form of the problem overcomes the singular behavior at the origin. The accuracy and applicability of our algorithm is examined by solving two singular models: (i) the second kind Lane–Emden equation used to model a thermal explosion in an infinite cylinder or a sphere and (ii) the nonlinear singular problem with Neumann boundary conditions.

A Single Period Production Inventory Model in Interval Environment with Price Revision

Abstract: In the present competitive market, in case of some products like smartphone, it is observed that after few days of launching a new product, a same type of product by some other company becomes available with a cheaper selling price or with more accessories within the same selling price. In this situation, customers normally go for the new product rejecting the older one. So always there is a possibility that some units of product remain unsold. This compels the company for a revision in selling price. This paper aims to develop such kind of production inventory models. Here demand of the product depends upon the selling price of the product, promotional effort and time. Now-a-days, specially for those kind of products, demand mainly depends on these three parameters in real life situation. When demand becomes zero, company sets a new selling price and the remaining products are depleted at a constant demand. So obviously it is going to be a single period inventory model. As a real life production process can not be hundred percent perfect, during the production process, some defective items are also produced which are reworked after the regular production process. Also, in order to reflect practical situation, inventory cost parameters are taken as interval numbers. As a result the corresponding optimization problem becomes interval valued which has been solved by using quantum-behaved particle swarm optimization technique. The main objective of this paper is to determine the optimal production run time that maximizes the total profit of the company.

A Note on Pathway Fractional Integral Formulas Associated with the Incomplete H-Functions

Abstract: Very recently Srivastava et al. (Russ J Math Phys 25(1):116–138, 2018) have introduced the incomplete H-functions and investigated their several interesting properties, for example, decomposition and reduction formulas, derivative formulas, and various integral transforms. They also pointed out potential applications of many of those incomplete special functions, which are specialized from the incomplete H-functions, involving (for example) probability theory. In this paper, we aim to establish two pathway fractional integral formulas involving the incomplete H-functions. Also our main results are indicated to reduce to yield some known identities. Further, among numerous special cases of our main results, some of them are explicitly demonstrated.

A New Integral Transform for Solving Higher Order Linear Ordinary Laguerre and Hermite Differential Equations

Abstract: In this work a new integral transform is introduced and applied to solve higher order linear ordinary Laguerre and Hermite differential equations. We compare present transform with other method such as Frobenius Method.

Stability of Integral Caputo-Type Boundary Value Problem with Noninstantaneous Impulses

Abstract: The modeling of a natural phenomena give soar to impulsive (instantaneous and noninstantaneous) fractional Caputo differential equations with boundary conditions. The behavior of the natural real world phenomena can be observed from the solutions of corresponding impulsive fractional Caputo differential equations with boundary conditions. Therefore, the existence, uniqueness and Ulam’s stability of the solutions of impulsive fractional Caputo differential equations are the most important concepts in fractional calculus. In this article, we take a noninstantaneous impulsive fractional Caputo differential equations with integral boundary conditions. The main objective of this article is, to study the existence, uniqueness and different types of Ulam’s stability for the solutions of fractional Caputo differential equations with noninstantaneous impulses and integral boundary conditions. At last, few examples are given to illustrate the new work.

Approximate Solution of Fractional Order Lane–Emden Type Differential Equation by Orthonormal Bernoulli’s Polynomials

Abstract: An approximate method based on orthonormal Bernoulli’s polynomials together with their operational matrices is applied for solving fractional order differential equations of Lane–Emden type. The preliminaries of fractional calculus are first presented. Operational matrices of fractional derivative and integer order derivative are constructed in this article. Convergence analysis of orthonormal Bernoulli’s polynomials is proposed here. By using this method, the fractional Lane–Emden differential equation converted into a system of algebraic equations by applying some suitable collocation points and this system can be simplified by an appropriate numerical method. Examples are illustrated to show the validity and applicability of the present method.

Efficient Iterative Method for Investigation of Convective–Radiative Porous Fin with Internal Heat Generation Under a Uniform Magnetic Field

Abstract: This paper is aimed at presenting an efficient iterative approach using Daftardar-Gejiji and Jafari method (DJM) for the analysis of thermal behaviour of convective–radiative porous fin with internal heat generation under a uniform magnetic field. The developed heat transfer models are used to investigate the effects of convective, radiative, and magnetic parameters on the thermal performance of the porous fin. From the study, we establish that increase in porosity, convective, radiative and magnetic parameters increase the heat transferred by the fin, which subsequently improves the fin efficiency. In addition, there is significant increase in heat transfer at the base of the fin whenever the thermal conductivity of the fin decreases. The result of DJM is validated by an established result of Adomian decomposition method, and compared with the results of numerical method using first-order Runge–Kutta with shooting method and homotopy analysis method. The comparison shows that Daftardar-Gejiji and Jafari’s method exhibits higher accuracy than the established two results.

A Highly Accurate Time–Space Pseudospectral Approximation and Stability Analysis of Two Dimensional Brusselator Model for Chemical Systems

Abstract: In this paper, the authors investigate the numerical solutions of two-dimensional reaction–diffusion equations with Neumann boundary conditions, known as Brusselator model, using Chebyshev pseudospectral method. The proposed methods are established in both time and space to approximate the solutions and prove the stability analysis of the equations. Higher order Chebyshev differential matrix is used in discretizing Brusselator model. The methods convert the Brusselator model into a system ofS algebraic equations, which are solved using Newton–Raphson method and obtained different types of patterns. Error of the proposed method is presented in terms of $$ L_{\infty }$$ and $$ L_{2}$$ error norms. In support of theoretical results, the methods are implemented on two problems and found highly accurate and stable approximations. The detailed comparison of the proposed method with various other methods are also given.

Selling Price Dependent Demand with Allowable Shortages Model Under Partially Backlogged—Deteriorating Items

Abstract: In this paper an inventory model for items with linear deterioration rate is studied. Demand depends upon the selling price of the commodity i.e. if selling price increases, the market demand decreases and vice versa. Shortages allowed and partially backlogged. There is neither repair nor replacement of deteriorated units occurring during the cycle. The ordering cost, purchasing cost, deteriorating cost, holding cost and the lost sale cost are considered in the inventory management. The model is solved analytically by minimizing the total inventory cost. Finally results are analyzed and demonstrated with illustrative numerical example. Sensitivity analysis is carried out to study the changing of different system parameters of this model.

Lie Symmetry Analysis and Some Exact Solutions of (2+1)-dimensional KdV-Burgers Equation

Abstract: In this work, the Lie group method is used to perform the similarity reduction and to obtain the exact solutions of the $$(2+1)$$ -dimensional KdV-Burgers equation. Similarity transformation method reduces $$(2+1)$$ -dimensional KdV-Burgers equation into $$(1+1)$$ -dimensional PDEs, later it reduces these PDEs into various ordinary differential equations and helps to find exact solutions of $$(2+1)$$ -dimensional KdV-Burgers equation. With the help of reduced equations, we have obtained the exact explicit solutions. Moreover, later by power series method, the exact analytic solutions of the KdV-Burgers equation are obtained.

Reduction of Kinetic Equations to Liénard-Levinson-Smith Form: Counting Limit Cycles

Abstract: We have presented an unified scheme to express a class of system of equations in two variables into a Lienard–Levinson–Smith (LLS) oscillator form. We have derived the condition for limit cycle with special reference to Rayleigh and Lienard systems for arbitrary polynomial functions of damping and restoring force. Krylov–Boguliubov (K–B) method is implemented to determine the maximum number of limit cycles admissible for a LLS oscillator atleast in the weak damping limit. Scheme is illustrated by a number of model systems with single cycle as well as the multiple cycle cases.

Numerical Solutions of Dissipative Natural Convective Flow from a Vertical Cone with Heat Absorption, Generation, MHD and Radiated Surface Heat Flux

Abstract: The laminar natural convective hydromagnetic viscous fluid flow induced by a cone under aspect of radiated heat flux with thermal radiation, heat absorption and generation is addressed here. The basic equations of conservation of momentum, mass and energy are utilized for the modeling of physical problem. The consequential expressions are worked out by using Crank–Nicholson approach. The implementation of this method leads to conversion of non-dimensional expressions into system of tri-diagonal expressions. The obtain numerical data is visualized for momentum, local-average shear stresses, rate of heat transportation and temperature for various constraints Pr, Δ, M, e and Rd with the help of graphical sketches. It is reported that the temperature of liquid is boost up with an enhancement in heat generation constraint. The larger Prandtl number corresponds to weaker temperature profiles. The average shear stress coefficient increase for higher radiation constraints and Prandtl number.

Interrelationships Between Marichev–Saigo–Maeda Fractional Integral Operators, the Laplace Transform and the $${\overline{H}}$$ H ¯ -Function

Abstract: In this article, we first evaluate the Laplace transform of Marichev–Saigo–Maeda (M–S–M) fractional integral operators whose kernel is the Appell function $$F_{3}$$ and point out its six special cases (Saigo, Erdelyi–Kober, Riemann–Liouville and Weyl fractional integral operators). Certain new and known results can be obtained as special cases of our key findings. Next, we find the image of $${\overline{H}}$$ -Function under the operators of our study. Some interesting special cases of our key result are also considered and demonstrated to be connected with certain existing ones.

Application of Fractional Operators in Modelling for Charge Carrier Transport in Amorphous Semiconductor with Multiple Trapping

Abstract: This paper is devoted to investigate the description of fractional space and time, mathematical modelling of charge carrier transport in disordered semiconductor. We propose the analytical approximate solutions of fractional drift diffusive equation that is under the influence of multiple trapping, by employing numerical technique the fractional reduced differential transform method (FRDTM). The present work gives wider description of FRDTM and provides approximate results in terms of convergent series. The approximate results through FRDTM, rendering that the proposed method is very simple, effective and reliable in use and may easily be applicable for solution of more general, non-linear fractional differential equations.

Study of Rayleigh–Bénard Convection of a Newtonian Nanoliquid in a High Porosity Medium Using Local Thermal Non-equilibrium Model

Abstract: In the paper we make linear and non-linear stability analyses of Rayleigh–Benard convection in a Newtonian, nanoliquid-saturated porous medium using local thermal non-equilibrium model (LTNE). The LTNE assumption results in advanced onset of convection and increase in heat transport when compared to that of local thermal equilibrium assumption. Free–free and rigid–rigid, isothermal boundaries are considered for investigation. The Galerkin method is used to obtain the critical eigen value. The influence of inter-phase heat transfer coefficient, ratio of thermal conductivities, Brinkman number, porous parameter on the onset of convection as well as on heat transport has been presented graphically and discussed in detail. The effect of increasing the value of porosity modified thermal conductivities advances the onset of convection and enhances the amount of heat transport whereas the remaining parameters have an opposing influence on both onset of convection as well as heat transport.

Effect of Magnetic Field on the Steady Viscous Fluid Flow Around a Semipermeable Spherical Particle

Abstract: An analytical solution is presented for the steady axisymmetrical Stokes flow of an electrically conducting viscous incompressible fluid through a partially permeable sphere subjected to a uniform transverse magnetic field. The considered flow is divided into two regions, outer viscous fluid region and inner semipermeable region, which are governed by modified Stokes and Darcy’s law respectively due to the presence of magnetic field in the flow regions. The boundary conditions used at the interface are continuity of normal component of velocity, vanishing of tangential component of velocity and continuity of pressure at the surface in contact with the fluid and the semipermeable sphere. An expression for drag force acting on the semipermeable sphere in presence of magnetic field is obtained. Enhancement of drag force exerted on the particle is seen on application of magnetic field. The effect of Hartmann numbers, permeability parameter on the hydrodynamic drag force were discussed. Some renowned results are obtained as the limiting cases.

Numerical Solution of Lane–Emden Type Equations Using Multilayer Perceptron Neural Network Method

Abstract: In this paper, we discuss the multi-layer perceptron artificial neural network technique for the solution of homogeneous and non-homogeneous Lane–Emden type differential equations. Our aim is to produce an optimal solution of Lane–Emden equations with less computation using multi-layer perceptron artificial neural network technique, comparatively other numerical techniques. Several test examples have been considered to determine the robustness of the given method. The results obtained prove that the given technique has the capability to develop into an effective approach for solving Lane–Emden type problems with less computation time and memory space.

Approximation of a System of Rational Functional Equations of Three Variables

Abstract: This study aims at substantiating the validity of stability results of a system of rational functional equations involving three variables connected with the Ulam stability theory of functional equations. There are some functional equations identified with real-time occurences. This system of functional equations is related with the intensive properties of substances such as density and volume.

Joint Dynamic Pricing and Inventory Control for Perishable Products Taking into Account Partial Backlogging and Inflation

Abstract: A joint dynamic pricing and inventory model by considering the inflation and time value of money where the product has a limited lifetime is developed. The selling price depends on product lifetime and customer demand rate depends on the selling price and time. Also, the shortages are considered and partially backlogged. The main goal is to determine the optimal initial selling price, the economic order quantity, and the optimal replenishment period simultaneously in order to maximize the total net present value of profit. We represent that for given any number of replenishment the function of profit is concave. In order to find the optimal solution, the presented heuristic algorithm is applied. To represent the algorithm and procedure of solution, a numerical example is solved. The results show that applying the proposed dynamic pricing approach leads to a significant improvement in profit’s retailer. Finally, sensitivity analysis of key parameters model is performed and some managerial insights are presented.