2024-10

Fractals (Issue : 1) (Volume : 2024)

This paper primarily lies in presenting existence and uniqueness analysis of boundary fractional difference equations of a special Riemann-Liouville operators' classes. To this end, we firstly develop Green’s function to the corresponding fractional boundary value problems and provide boundary conditions to find the constants. Then we study the existence of solutions, and we examine the bounded of their solutions. Eventually, two numerical examples are given to demonstrate the efficiency and uniqueness behavior of the boundary value problem. Furthermore, such fractional problems can typically be converted to a direct fractional problem with certain boundary or initial conditions in transport in porous media.

2024-09

Mathematics (Issue : 12) (Volume : 18)

Metamaterials have emerged as a focal point in contemporary science and technology due to their ability to drive significant innovations. These engineered materials are specifically designed to couple the phenomena of different physical natures, thereby influencing processes through mechanical or thermal effects. While much of the recent research has concentrated on frequency conversion into electromagnetic waves, the field of acoustic frequency conversion still faces considerable technical challenges. To overcome these hurdles, researchers are developing metamaterials with customized acoustic properties. A key equation for modeling nonlinear acoustic wave phenomena is the dissipative Westervelt equation. This study investigates analytical solutions using ansatz-based methods combined with Lie symmetries. The approach presented here provides a versatile framework that is applicable to a wide range of fields in metamaterial design.

2024-09

Mathematics (Issue : 12) (Volume : 18)

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi-𝛽 -Banach spaces and multi-Banach spaces. Additionally, the study explores the stability of the equation in various extended Banach spaces and provides a specific example illustrating the absence of stability in certain cases.

2024-09

Mathematics (Issue : 12) (Volume : 18)

This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the corresponding fractional boundary value problems will be reconstructed, and homogeneous boundary conditions are used to find the unknown constants. Next, the existence of solutions will be studied depending on the fixed-point theorems on the constructed Green’s function. The uniqueness of the problem is also derived via Lipschitz constant conditions.

2024-09

Axioms (Issue : 13) (Volume : 10)

This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian and corresponding equations of motion are derived. The findings demonstrate that both Hamiltonian and Euler–Lagrange formulations yield equivalent results. This study highlights the Hamiltonian approach as a valuable alternative for understanding the dynamics of second-order systems, validated through a specific application within generalized electrodynamics. The novelty of the research lies in developing advanced theoretical models through Hamiltonian formalism for continuous systems with second-order derivatives. The research employs an alternative method to the Euler–Lagrange formulas by applying Dirac’s theory to study the generalized Podolsky electrodynamics, contributing to a better understanding of complex continuous systems.

2024-08

Axioms (Issue : 8) (Volume : 13)

In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence and conditional stability. The accuracy of the method is demonstrated with 3D mesh plots, while the effects of time and fractional order are shown in 2D plots. Comparative evaluations with the cubic B-spline collocation method are provided. To illustrate the suitability and effectiveness of the proposed method, two examples are tested, with the results are evaluated using 𝐿2 and 𝐿∞ norms.

2024-08

Axioms (Issue : 8) (Volume : 13)

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size 𝑝0 tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method.

2024-07

Mathematical and Computer Modelling of Dynamical Systems (Issue : 1) (Volume : 30)

In this study, the sequential operator of mixed order is analysed on the domain (μ2,μ1)∈(0,1)×(0,1) with 1<μ2+μ1<2, Then, the positivity of the nabla operator is obtained analytically on a finite time scale under some conditions. As a consequence, our analytical results are introduced on a set, named Em,ζ, on which the monotonicity analysis is obtained. Due to the complicatedness of the set Em,ζ several numerical simulations so are applied to estimate the structure of this set and they are provided by means of heat maps.

2024-06

AIMS Mathematics (Issue : 7) (Volume : 9)

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $L_2$ and $L_\infty$ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.

2024-06

PLOS ONE (Issue : 19) (Volume : 6)

This research presents a novel conformable-Caputo fractional non-polynomial spline method for solving the time-fractional Korteweg-de Vries (KdV) equation. Emphasizing numerical analysis and algorithm development, the method offers enhanced precision and modeling capabilities. Evaluation via the Von Neumann method demonstrates unconditional stability within defined parameters. Comparative analysis, supported by contour and 2D/3D graphs, validates the method’s accuracy and efficiency against existing approaches. Quantitative assessment using L2 and L∞ error norms confirm its superiority. In conclusion, the study proposes a robust solution for the time-fractional KdV equation.

2024-05

Symmetry (Issue : 5) (Volume : 16)

Recently, nonlinear fractional models have become increasingly important for describing phenomena occurring in science and engineering fields, especially those including symmetric kernels. In the current article, we examine two reliable methods for solving fractional coupled nonlinear Schrödinger models. These methods are known as the Sardar-subequation technique (SSET) and the improved generalized tanh-function technique (IGTHFT). Numerous novel soliton solutions are computed using different formats, such as periodic, bell-shaped, dark, and combination single bright along with kink, periodic, and single soliton solutions. Additionally, single solitary wave, multi-wave, and periodic kink combined solutions are evaluated. The behavioral traits of the retrieved solutions are illustrated by certain distinctive two-dimensional, three-dimensional, and contour graphs. The results are encouraging, since they show that the suggested methods are trustworthy, consistent, and efficient in finding accurate solutions to the various challenging nonlinear problems that have recently surfaced in applied sciences, engineering, and nonlinear optics.

2024-04

Symmetry (Issue : 4) (Volume : 16)

There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that are related to interval-valued cr-log-h-convex functions, and some known results are recaptured. To support our main results, we offer three examples and two applications related to modified Bessel functions and special means as well.

2024-03

Symmetry (Issue : 16) (Volume : 3)

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green’s function kernel. In the deterministic case, Green’s function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type.

2023-11

Mathematics and Computers in Simulation (Issue : 2024) (Volume : 218)

This research introduces the fractional non-polynomial spline method as a novel scheme for solving the time-fractional Korteweg-de Vries (KdV) equation. The study focuses on numerical analysis and algorithm development for simulation purposes. The proposed method involves the construction of a fractional non-polynomial spline to estimate the equation's solution, offering improved precision and modelling capabilities compared to existing approaches. To assess the stability of the proposed approach, the von Neumann method is employed, demonstrating its unconditional stability within a specific parameter range. To validate the effectiveness of our numerical analysis and simulation algorithm, contour, 2D, and 3D graphs are utilized to compare the solution obtained through our method with an analytical solution. Through rigorous comparative analysis with previous works, the superiority of our approach in terms of accuracy and efficiency is demonstrated. Norm error calculations, specifically the (L_2 and L_\infinity) error norms, provide a quantitative assessment of the accuracy and reliability of our proposed scheme.

2023-09

Mathematics (Issue : 17) (Volume : 11)

Abstract This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (𝐿2 and 𝐿∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

2023-08

Ain Shams Engineering Journal (Issue : 11) (Volume : 14)

The article introduces a novel and developed a numerical technique that solves nonlinear time fractional differential equations. It combines finite difference and non-polynomial spline methods while using conformable descriptions of fractional derivatives. The strategy has proven to be useful to analyzing intricate datasets across various domains, including finance, science, and engineering. The applicability and validity of the approach are demonstrated through numerical examples. The study assessed the approach's stability using the Fourier method and found it to be unconditionally stable for a specific parameter range. The proposed method has been analyzed for convergence, and the analysis reveals that it exhibits a convergence order of six. The paper also includes graphs comparing the present solution to an analytical one. The method is tested with some examples of the model that have a wide range of applications in fields such as biology, ecology, and physics called the Burgers-Fisher equations and compared to previous approaches to demonstrate its effectiveness and applicability. The research evaluated the approach's accuracy and efficiency using the (L2 and L_inf) norm errors. Also, the effects of time and fractional derivatives have been studied, and to the best of our knowledge, it has not been previously published in any academic or industry publication.

2023-05

A new numerical scheme non-polynomial spline for solving generalized time fractional Fisher equation (Issue : 5) (Volume : 44)

In this paper, a novel numerical scheme is developed using a new construct by non-polynomial spline for solving the time fractional Generalize Fisher equation. The proposed models represent bacteria, epidemics, Brownian motion, kinetics of chemicals and fuzzy systems. The basic concept of the new approach is constructing a non-polynomial spline with different non-polynomial trigonometric and exponential functions to solve fractional differential equations. The investigated method is demonstrated theoretically to be unconditionally stable. Furthermore, the truncation error is analyzed to determine the or-der of convergence of the proposed technique. The presented method was tested in some examples and compared graphically with analytical solutions for showing the applicability and effectiveness of the developed numerical scheme. In addition, the present method is compared by norm error with the cubic B-spline method to validate the efficiency and accuracy of the presented algorithm. The outcome of the study reveals that the developed construct is suitable and reliable for solving nonlinear fractional differential equations.

2022-11

AIP Advances (Issue : 11) (Volume : 12)

In a recent study, we investigate the Burgers–Fisher equation through a developed scheme, namely, the non-polynomial spline fractional continuity method. The proposed models represent nonlinear optics, chemical physics, gas dynamics, and heat conduction. The basic concept of the new approach is constructing a non-polynomial spline with a fractional continuity equation instead of a natural derivative. Furthermore, the truncation error is analyzed to determine the order of convergence for the proposed scheme, and we presented theoretically the stability of the developed scheme using the von Neumann method. One might easily conclude that the new scheme is quite successful and effective in obtaining the numerical solutions of the time partial/fractional partial differential equations. In addition, we plotted contour, 2D, and 3D graphs for some reported solutions to compare the presented solution with an exact solution. The investigated method was tested in some examples and compared to previous solutions for showing the applicability and effectiveness of the developed numerical scheme. The absolute and norm errors L2 and L∞ has calculated to validate the accuracy and efficiency of the presented scheme. To our knowledge, all obtained solutions in this research paper are novel and not published beforehand.

2021-10

AIP Advances (Issue : 11) (Volume : 10)

In this work, a modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota-Satsuma coupled KdV equation with time as a series solution. Comparison of the analytical approximate solution with the exact solution concluded that the present method is an important addition for analyzing a system of partial differential equations that have a strong nonlinear term. We also represented graphically and discussed the effect of initial condition parameters and reaction of time on the model.

2019-12

Pramana (Issue : 6) (Volume : 93)

In this article, the effect of thermo-diffusion and diffusion-thermo on hyperbolic tangent magnetised nanofluid with Hall current past a nonlinear porous stretching surface has been analysed numerically. The impact of thermal slip and chemical reaction are also examined in our current analysis. Runge–Kutta–Merson method and shooting method have been successfully employed to obtain numerical results for the governing nonlinear differential equations. The impact of Hartmann number, Hall parameter, porosity parameter, fluid parameter, Weissenberg number, Richardson number, concentration buoyancy parameter, Schmidt number, Dufour parameter, Soret number, Prandtl number, chemical reaction parameter, and power-law exponent are discussed and demonstrated graphically for the flow phenomena. Furthermore, the description for Sherwood number, rate of shear stress, and Nusselt number are displayed using tables against all the pertinent parameters. A detailed numerical comparison for the power-law exponent and Prandtl number has been elaborated via tables.

2019-03

Heat Transfer Research (Issue : 50) (Volume : 7)

In this article, the magnetohydrodynamic (MHD) thermal boundary layer of a Carreau fl ow of Cu–water nanofluids over an exponentially permeable stretching thin plate is investigated numerically. Internal heat source/sink is also taken into account. After gaining the system of leading equations, the appropriate transformations have been first employed to come across the fitting parallel conversions to alter the central governing equations into a suit of ODEs and then the renovated system of ODE along with appropriate boundary conditions is numerically solved by the shooting method with fourth-order Runge-Kutta technique. The consequences of the relevant factors of physical parameters on velocity and temperature of merging water (H2O) and nanoparticles (Cu) have been exemplified through graphs.

2018-12

Journal of Magnetics (Issue : 23) (Volume : 4)

Effects of the uniform magnetic field on aqueous magneto-Nanofluid confined in a porous domain with wall stretching/shrinking non-linearly is analyzed via this communication. The problem is modeled using continuity, momentum and energy equation along with linear thermal radiation. The effects of physical quantities are observed for Cu, Al2O3, TiO2 and Ag particles in water. The coupled boundary layer PDE’s are reduced into the system of ODEs by utilizing similarity transformation and solved using shooting and Runga-Kutta fourth order technique. Stability of the obtained results are also analyzed. The results are displayed through graphs. It is observed that the momentum boundary layer is thicker when silver particles are introduced in water. Whereas, temperature profile has the minimum value for silver nanoparticles and maximum for Titanium dioxide. Also, in case of shrinking sheet dual solutions are obtained along with smallest Eigen values.

2018-04

Arab Journal of Basic and Applied Sciences (Issue : 1) (Volume : 25)

Three-dimensional analysis of condensation nanofluid film on an inclined rotating disk by efficient analytical methods

2017-12

International Journal of Applied and Computational Mathematics (Issue : 3) (Volume : 7)

In this paper, three dimensional incompressible Casson fluid flow past a linear stretching porous plate under the magnetic field effect is analyzed using differential transformation method (DTM) and numerical method. For increasing the accuracy of DTM, Padé approximation is applied. Comparison betweenDTM-Padé and numerical method shows that Padé with order [15,15] can be an exact and high efficiency procedure for solving these kinds of problems. The influence of the Casson fluid parameter (β), Prandtl number (Pr), magnetic parameter (M), stretching parameter (c) and porous parameter (λ) on non-dimensional temperature and velocity profiles are investigated. The results indicated for the Casson fluid flow that increasing the Hartmann numbermake a decrease in velocity boundary layer thicknesses.

2017-04

Nonlinear Dynamics (Volume : 89)

In this paper, strongly nonlinear partial differential equations termed the modified Kawahara equations are investigated analytically using residual power series method, a modern and effective method. The method supplies good accuracy for analytical solutions when compared to exact solutions. By means of an illustrative example we show that the present technique performs better than other methods for solving nonlinear equations. The action time and influence of term parameters of terms are shown graphically.

2017-03

Journal of Mathematics and Computer Science-JMCS (Issue : 1) (Volume : 17)

The problem of MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet is investigated analytically. Governing equations are reduced to a set of nonlinear ordinary differential equations using the similarity transformations, and solved via an efficient and suitable mathematical technique, named the differential transform method (DTM), in the form of convergent series, by applying Pad´e approximation. The results are compared with the results obtained by the shooting method of MATHEMATICA and with the fourth-order Runge-Kutta-Fehlberg results. The results of DTM-Padé are closer to numerical solutions than the results of DTM are. A comparison of our results with existing published results shows good agreement between them. Suitability end effectiveness of our method are illustrated graphically for various parameters. Moreover, it is also observed that the Casson fluid parameter, stretching parameter, Hartmann number and porosity parameter increase with increment in the velocity profiles.

2017-02

Journal of the Association of Arab Universities for Basic and Applied Sciences (Volume : 10)

In this paper, we practiced relatively new, analytical method known as the variational homotopy perturbation method for solving Klein–Gordon and sine-Gordon equations. To present the present method’s effectiveness many examples are given. In this study, we compare numerical results with the exact solutions, the Adomian decomposition method (ADM), the variational iteration method (VIM), homotopy perturbation method (HPM), modified Adomian decomposition method (MADM), and differential transform method (DTM). The results reveal that the VHPM is very effective.

2017-02

Journal of Mathematics and Computer Science-JMCS (Issue : 1) (Volume : 17)

In this paper, Differential Transformation Method (DTM) is applied on governing equations of heat and fluid flow for a nanofluid over a horizontal flat plate. After obtaining the governing equations and solving them by DTM, the accuracy of results is examined by fourth order Runge-kutta numerical method. Due to infinite boundary condition for the stretching plate, outcomes need to an improvement method to be converged. For this aim, Padé approximation is applied on the obtained results which [10,10] Padé order had the best accuracy compared to numerical method. The influence of relevant parameters such as the transpiration parameter on temperature and nanoparticle concentration profile is discussed and it is concluded that by increasing this parameter, nanoparticles concentration over the plate decrease due to more fluid penetration from pores and this is the main reason of lower thermal boundary layer caused by fewer nanoparticles over the plate.

2017-01

International Journal of ADVANCED AND APPLIED SCIENCES (Issue : 4) (Volume : 1)

In this research, we analyze heat transfer of MHD boundary layer flow of Casson fluid. Strong nonlinear ordinary differential equations are solved numerical using Shooting method with fourth order Runge-Kutta (RK4) integration technique. Variations of interesting different parameters on the velocity are showed graphically.

2017-01

Cogent Mathematics (Issue : 4) (Volume : 1)

In this paper, a residual power series method (RPSM) is combining Taylor’s formula series with residual error function, and is investigated to find a novel analytical solution of the coupled strong system nonlinear Boussinesq-Burgers equations according to the time. Analytical solution was purposed to find approximate solutions by RPSM and compared with the exact solutions and approximate solutions obtained by the homotopy perturbation method and optimal homotopy asymptotic method at different time and concluded that the present results are more accurate and efficient than analytical methods studied. Then, analytical simulations of the results are studied graphically through representations for action of time and accuracy of method.

2016-04

International Journal of Pure and Applied Mathematics (Issue : 107) (Volume : 2)

In this article, the flow of liquid film over an unsteady elastic stretching surface is analyzed. Similarity transformations are used to transform the governing equations to a nonlinear ordinary differential equation. The differential equation reformulated to system of Voltera integral equations and solved analytically using the new technique of numerical solution called homotopy perturbation method. The results of the proposed method are compared with previously published work and the results are found to be in an excellent agreement. Also, discussed and presented graphically the effects of various parameters Darcy number and unsteadiness parameter.

2015-10

American Journal of Computational Mathematics (Volume : 5)

In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.

2015-05

Applied Mathematics (Issue : 6) (Volume : 4)

In this article, the application of variational homotopy perturbation method is applied to solve Benjamin-Bona-Mahony equation. Then, we obtain the numerical solution of BBM equation using the initial condition. Comparison with Adomian's decomposition method, homotopy perturbation method, and with the exact solution shows that VHPM is more effective and accurate than ADM and HPM, and is reliable and manageable for this type of equation.

2015-01

British Journal of Mathematics & Computer Science (Issue : 11) (Volume : 2)

In this paper, a competition model of a reaction diffusion system with self-diffusion has been studied using homotopy perturbation method, variational iteration method and Finite element method FEM (COMSOL package). The traveling wave solutions for this system are found and compared numerically. It was shown that the competition will lead at the end of the winning of one species. The effect of self diffusion is shown in the dispersing of traveling wave solution. Also, it was shown that the solution of finite element method and homotopy perturbation method are convergent to each other compared to the variation iteration method.

2014-06

International Journal of Applied Mathematical Research (Issue : 3) (Volume : 3)

In this study, the approximate solutions for the Kuramoto-Sivashinsky equation by using the Variational Iteration Method (VIM) are obtained. Comparisons with the exact solutions and the solutions obtained by the Homotopy Perturbation Method (HPM), the numerical example show that the Variational Iteration Method (VIM) is accurate and effective and suitable for this kind of problem.

2014-01

IOSR Journal of Engineering (Issue : 10) (Volume : 1)

The approximate solutions for the Kuramoto –Sivashinsky Equation are obtained by using the Adomian Decomposition method (ADM). The numerical example show that the approximate solution comparing with the exact solution is accurate and effective and suitable for this kind of problem.

2013-12

IOSR Journal of Engineering (Issue : 3) (Volume : 12)

The approximate solutions for the Kuramoto –Sivashinsky Equation are obtained by using the homotopy perturbation method (HPM). The numerical example show that the approximate solution comparing with the exact solution is accurate and effective and suitable for this kind of problem.