- I'm a Prof. Applied Mathematics, published 300 papers.My Scopus h-index: 34 Citation: 4319edit
Research Interests:
The present paper is aimed at studying the effect of gravity o n the general model of the equations of the generalized therm omicrostretch for a homogeneous isotropic elastic half-spa ce solid whose surface is subjected to a Mode-I crack... more
The present paper is aimed at studying the effect of gravity o n the general model of the equations of the generalized therm omicrostretch for a homogeneous isotropic elastic half-spa ce solid whose surface is subjected to a Mode-I crack problem . The problem is in the context of the Green and Naghdi theory (GN). The norm al ode analysis is used to obtain the exact expressions for t he displacement components, the force stresses, the temperat ure, the couple stresses and the microstress distribution. The variations of the considered variables through the horizontal distance are i llustrated graphically. Comparisons are made with the resu lts in the presence and absence of gravity with two cases: Case (1) for the genera lized micropolar thermoelasticity elastic medium (withou t microstretch constants) between the both types (II, III). Case (2) for the generalized micropolar thermoelasticity elastic medium ( without micropolar constants) between the both types II and III.
The aim of this paper is to introduce the Green-Naghdi (G-N) theory of type III (with energy dissipation) to study the effect of thermal loading due to laser pulse on generalized micropolar thermoelastic homogeneous isotropic medium in... more
The aim of this paper is to introduce the Green-Naghdi (G-N) theory of type III (with energy dissipation) to study the effect of thermal loading due to laser pulse on generalized micropolar thermoelastic homogeneous isotropic medium in three dimensions. The normal mode analysis technique is used to solve the resulting non-dimensional equations of the problem. Numerical results for the displacement, thermal stress, strain, temperature, couple stresses and micro-rotation distributions are represented graphically to display the effect of the laser pulse on the resulting quantities. Comparisons are made within the theory in the presence and absence of the laser pulse.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
The purpose of this paper is to study the thermoviscoelastic interactions in a homogeneous, isotropic semi-infinite solid under two-temperature theory with heat source. The Kelvin-Voigt model of linear viscoelasticity which describes the... more
The purpose of this paper is to study the thermoviscoelastic interactions in a homogeneous, isotropic semi-infinite solid under two-temperature theory with heat source. The Kelvin-Voigt model of linear viscoelasticity which describes the viscoelastic nature of the material is used. The bounding plane surface of the medium is subjected to a non-Gaussian laser pulse. The generalized thermoelasticity theory with dual phase lags model is used to solve this problem. Laplace transform technique is used to obtain the general solution for a suitable set of boundary conditions. Some comparisons have been shown in figures to estimate the effects of the phase lags, viscosity, temperature discrepancy, laser-pulse and the laser intensity parameters on all the studied fields. A comparison was also made with the results obtained in the case of one temperature thermoelasticity theory.
Research Interests:
The dual-phase-lag (DPL) model is applied to study the influence of rotation on a two-dimensional micro-elongated thermoelastic medium problem. Mechanical force along with the layer of the elastic half-space interface and micro-elongated... more
The dual-phase-lag (DPL) model is applied to study the influence of rotation on a two-dimensional micro-elongated thermoelastic medium problem. Mechanical force along with the layer of the elastic half-space interface and micro-elongated thermoelastic half-space is applied. The analytic expressions for displacement component, temperature distribution, micro-elongational scalar and stress components have been derived and represented graphically. The rotation has been studied in the presence DPL model and Lord-Shulman theory.
This paper aims to investigate the effect of rotation on a micropolar thermoelastic medium with voids problem. The problem is assessed according to three-phase-lag model. The normal mode analysis used to obtain the analytical expressions... more
This paper aims to investigate the effect of rotation on a micropolar thermoelastic medium with voids problem. The problem is assessed according to three-phase-lag model. The normal mode analysis used to obtain the analytical expressions of the considered variables. The non-dimensional displacement, temperature, Micro rotation, the change in the volume fraction field, and stress of the material are obtained and illustrated graphically. Comparisons are made with the results predicted by two theories; namely three- phase-lag model (3PHL) and Green-Naghdi theory of type III (G-N III). The considered variables were plotted for different values of the rotation parameter, the phase-lag of heat flux and the phase-lag of temperature. The numerical results reveal that the rotation and the phase-lag times significantly influence the distribution of the field quantities. Some particular cases of interest are deduced from the present investigation.
Research Interests:
A new model of equations of generalized thermoelasticity for an isotropic medium with mechanical properties that are dependent on temperature is established. The present problem is a generalization of the three-phase-lag model, Lord and... more
A new model of equations of generalized thermoelasticity for an isotropic medium with mechanical properties that are dependent on temperature is established. The present problem is a generalization of the three-phase-lag model, Lord and Shulman's coupled theory. The elasticity modulus is a reference temperature function which is linear. Analytical expressions of the considered variables are obtained by using the Laplace–Fourier transforms technique. The results are analysed in a deeper manner by comparing them with unique cases of absence of the magnetic field, temperature-dependent properties of the body, and two types of mechanical loads. The most significant points are highlighted.
Research Interests:
Female, 82 FINAL DIAGNOSIS: Achalasia Symptoms: Nocturnal regurgtation • weight loss - Clinical Procedure: Esophageal stenting Specialty: Gastroenterology • Hepatology Objective: Unusual or unexpected effect of treatment. Pneumatic... more
Female, 82 FINAL DIAGNOSIS: Achalasia Symptoms: Nocturnal regurgtation • weight loss - Clinical Procedure: Esophageal stenting Specialty: Gastroenterology • Hepatology Objective: Unusual or unexpected effect of treatment. Pneumatic dilatation is one of the most effective methods for treating achalasia. Esophageal perforation is the most serious complication after pneumatic dilatation and has been reported to occur in the range of 1 to 4.3%. The appropriate management of esophageal perforation can range from conservative medical treatment to surgical intervention. We report a case of an 82-year-old male who had an 8 month history of dysphagia for solid and liquids, a 10 lb weight loss and nocturnal regurgitation. The diagnosis of achalasia was established by endoscopic; barium and manometric criteria. He underwent a pneumatic dilation with a 30 mm Rigiflex balloon. A confined or limited esophageal perforation projecting into the mediastinum and located 1-2 cm above the diaphragm was ...
Research Interests:
Introduction. Colorectal anastomotic leak or stricture is a dreaded complication leading to significant morbidity and mortality. The novel use of self-expandable metal stents (SEMS) in the management of postoperative colorectal... more
Introduction. Colorectal anastomotic leak or stricture is a dreaded complication leading to significant morbidity and mortality. The novel use of self-expandable metal stents (SEMS) in the management of postoperative colorectal anastomotic leaks or strictures can avoid surgical reintervention. Methods. Retrospective study with particular attention to the indications, operative or postoperative complications, and clinical outcomes of SEMS placement for patients with either a colorectal anastomotic stricture or leak. Results. Eight patients had SEMS (WallFlex stent) for the management of postoperative colorectal anastomotic leak or stricture. Five had a colorectal anastomotic stricture and 3 had a colorectal anastomotic leak. Complete resolution of the anastomotic stricture or leak was achieved in all patients. Three had recurrence of the anastomotic stricture on 3-month flexible sigmoidoscopy follow-up after the initial stent was removed. Two of these patients had a stricture that wa...
Research Interests:
Calculating the adenoma detection rate (ADR) is a complex process in contrast to the polyp detection rate (PDR) that can be easily calculated. The average adenoma to polyp detection rate quotient (APDRQ) was proposed as a conversion... more
Calculating the adenoma detection rate (ADR) is a complex process in contrast to the polyp detection rate (PDR) that can be easily calculated. The average adenoma to polyp detection rate quotient (APDRQ) was proposed as a conversion factor to estimate the ADR for individual endoscopists from the endoscopist's PDR. However, this conversion factor was not validated in different practice settings. To validate the use of the proposed conversion factor in a practice setting with a predominantly Hispanic population. We conducted a retrospective, cross-sectional study (December 2007 to November 2012) of screening colonoscopies at a university practice setting with an 86.9% Hispanic population. The actual ADR and PDR were calculated for all endoscopists. The weighted average of ADR to PDR ratio for each endoscopist was used to obtain APDRQ. The APDRQ was used as a conversion multiplier to estimate each endoscopist's ADR using the single endoscopist's PDR. A total of 2148 screening colonoscopies were included. The average PDR for the whole group was 36.9% (range, 11% to 49%). The actual ADR was estimated as 25.5% (range, 11% to 37%). The average APDRQ for our group was 0.68. The estimated ADR was 25.48% (range, 8% to 33%). There was a high correlation between actual ADR and the estimated ADR (Pearson correlation=0.92). In a practice setting with a predominantly Hispanic population, a conversion factor can be used to estimate ADR from PDR providing a high degree of correlation with the actual ADR.
Research Interests:
A model of the equations of two-dimensional problems is studied in a half space, whose surface in a medium free of micropolar thermoelastic possesses cubic symmetry as a result of inclined load. There acts an initial magnetic parallel to... more
A model of the equations of two-dimensional problems is studied in a half space, whose surface in a medium free of micropolar thermoelastic possesses cubic symmetry as a result of inclined load. There acts an initial magnetic parallel to the plane boundary of the half-space. The inclined load is assumed to be a linear combination of a normal load and a tangential load. The formulation is performed in the context of the Lord-Shulman and Green-Lindsay theories, as well as the classical dynamical coupled theory. Comparisons are made with the results in the presence of a magnetic.
Research Interests:
A model of the equations of two-dimensional problems is studied in a half space, whose surface in a medium free of micropolar thermoelastic possesses cubic symmetry as a result of inclined load. There acts an initial magnetic parallel to... more
A model of the equations of two-dimensional problems is studied in a half space, whose surface in a medium free of micropolar thermoelastic possesses cubic symmetry as a result of inclined load. There acts an initial magnetic parallel to the plane boundary of the half-space. The inclined load is assumed to be a linear combination of a normal load and a tangential load. The formulation is performed in the context of the Lord-Shulman and Green-Lindsay theories, as well as the classical dynamical coupled theory. Comparisons are made with the results in the presence of a magnetic.
A new simple well-behaved definition of the fractional derivative termed as conformable fractional derivative and introducing a geometrical approach of fractional derivatives, non-integral order initial value problems are an attempt to... more
A new simple well-behaved definition of the fractional derivative termed as conformable fractional derivative and introducing a geometrical approach of fractional derivatives, non-integral order initial value problems are an attempt to solve in this article. Based on the geometrical interpretation of the fractional derivatives, the solution curve is approximated numerically. Two special phenomena are employed for concave upward and downward curves. In order to obtain the solution of fractional order differential equation (FDE) with the integer-order initial condition, some new criteria on fractional derivatives are proposed.
Research Interests:
In this article, the variational iteration method is used to solve an ordinary differential equation of N-order boundary value problems. We solve this problem by changing the problem to a system of two integral-differential equations... more
In this article, the variational iteration method is used to solve an ordinary differential equation of N-order boundary value problems. We solve this problem by changing the problem to a system of two integral-differential equations (2,5,16,18) and using the variational it- eration method (6-10,12). By giving three examples as ninth-order, tenth-order and twelve-order and comparing with the exact solution, the efficiency of the method will be shown.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
In the present paper, the two different theories (coupled theory and Green–Lindsay theory with two relaxation times) are applied to study the deformation of a generalized piezo-thermoelastic rotating medium under the influence of magnetic... more
In the present paper, the two different theories (coupled theory and Green–Lindsay theory with two relaxation times) are applied to study the deformation of a generalized piezo-thermoelastic rotating medium under the influence of magnetic field. The normal mode analysis is used to obtain the expressions for the displacement components, the temperature, the stress, the strain components, the electric potential and the electric displacements. Numerical results for the field quantities are given in the physical domain and illustrated graphically. Comparisons are made with the results predicted by coupled and Green–Lindsay theories in the presence and absence of rotation as well as magnetic field.
Research Interests:
Abstract The purpose of the present paper is to study the gravitational effect in a homogeneous isotropic semiconducting medium subjected to an internal heat source, based on Lord–Shulman theory. The problem is formulated in the... more
Abstract The purpose of the present paper is to study the gravitational effect in a homogeneous isotropic semiconducting medium subjected to an internal heat source, based on Lord–Shulman theory. The problem is formulated in the dimensionless form and then solved analytically using the normal mode analysis method. The physical quantities are obtained and tested by a numerical study using the parameters of silicon as a target and presented graphically. The distribution of these quantities is represented graphically in the presence and absence of the internal heat source as well as gravity.
Research Interests:
Research Interests:
Abstract.In the present paper, two different theories (coupled theory and Green-Lindsay theory with two relaxation times) are applied to study the deformation of a generalized piezothermoelastic rotating medium under the influence of... more
Abstract.In the present paper, two different theories (coupled theory and Green-Lindsay theory with two relaxation times) are applied to study the deformation of a generalized piezothermoelastic rotating medium under the influence of gravity. The normal mode analysis is used to obtain the expressions for the displacement components, the temperature, the stress, the stress components, the electric potential and the electric displacements. Numerical results for the field quantities are given in the physical domain and illustrated graphically. Comparisons are made with the results predicted by coupled and Green-Lindsay theories in the presence and absence of rotation as well as of gravity.
Research Interests:
Research Interests:
The model of the two-dimensional equations of generalized magneto-thermoelasticity with two relaxation times in a perfectly conducting medium is established. The normal mode analysis is used to obtain the exact expressions for the... more
The model of the two-dimensional equations of generalized magneto-thermoelasticity with two relaxation times in a perfectly conducting medium is established. The normal mode analysis is used to obtain the exact expressions for the temperature distribution, thermal ...
Research Interests: Mechanical Engineering, Civil Engineering, Applied Mathematics, Modeling, Heat Transfer, and 14 moreModal Analysis, Magnetic field, Thermal Stress, Thermal, Energy Dissipation, Temperature Distribution, Engineering Science, Thermal Stresses, Time Dependent, Numerical computation, Boundary Condition, Normal Mode Analysis, Relaxation Time, and plane waves
Research Interests:
A two-dimensional coupled problem in generalized thermoelasticity for rotating media under the temperature dependent properties is studied. The problem is in the context of the Lord-Shulman's theory with one relaxation time. The... more
A two-dimensional coupled problem in generalized thermoelasticity for rotating media under the temperature dependent properties is studied. The problem is in the context of the Lord-Shulman's theory with one relaxation time. The normal mode analysis is used to obtain the expressions for the temperature distribution, displacement components and thermal stresses. The resulting formulation is applied to two different problems. The first concerns the case of a heat punch moving across the surface of a semi-infinite thermoelastic half-space subjected to appropriate boundary conditions. The second deals with a thick plate subject to a time-dependent heat source on each face. Numerical results are illustrated graphically for each problem considered. Comparisons are made with the results obtained predicted by the two theories in case of absence of rotation.
Research Interests:
Research Interests:
In this paper, we established the generalized thermoelasticity phenomenon in an isotropic elastic medium considering the electromagnetic field, rotation and two-temperature. Three theories of generalized thermoelasticity have been... more
In this paper, we established the generalized thermoelasticity phenomenon in an isotropic elastic medium considering the electromagnetic field, rotation and two-temperature. Three theories of generalized thermoelasticity have been applied: Lord-Shulman (one relaxation time), Green-Lindsay (two relaxation times), as well as the coupled theory. We discussed some particular cases in the context of the wave propagation phenomenon in thermoelasticity. From solving the fundamental equations, we arrived that there are three waves: P-, T- and SV-waves that we calculated their velocities. The boundary conditions for mechanical stress and Maxwell's stress and thermal insulated or isothermal have been applied to determine the amplitudes ratios (reflection coefficients) for P-, T - and SV waves. Some utilitarian aspects are obtained from the reflection coefficients, presented graphically, and the new conclusions have been presented. Comparisons are made for the results predicted by differen...
Research Interests:
A new theory of the generalized thermoelasticity reinforcement has been constructed by taking into account the deformation of a micropolar generalized thermoelastic medium with voids. A general model of the equations of the formulation in... more
A new theory of the generalized thermoelasticity reinforcement has been constructed by taking into account the deformation of a micropolar generalized thermoelastic medium with voids. A general model of the equations of the formulation in the context of the Lord-Shulman (L-S) theory which includes one relaxation time and Green-Lindsay (G-L) theory with two relaxation times, as well as the classical dynamical coupled (CD) theory, are applied to study the influence of reinforcement on the total deformation of a micropolar thermoelastic medium with voids. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses and temperature. The variations of the considered variables with the horizontal distance are illustrated graphically. Comparisons are made with the results in the three theories in the presence and absence of fiber- reinforcing.
Research Interests:
In this paper, the problem of the generalized thermoelastic medium for three different theories under the effect of a laser pulse and two-temperature is investigated. The Lord–Shulman (L-S), Green-Naghdi of type III (G-N III) and... more
In this paper, the problem of the generalized thermoelastic medium for three different theories under the effect of a laser pulse and two-temperature is investigated. The Lord–Shulman (L-S), Green-Naghdi of type III (G-N III) and three-phase-lag (3PHL) theories are discussed with two-temperature. The normal mode analysis is used to obtain the analytical expressions of the displacement components, force stress, thermodynamic temperature and conductive temperature. The numerical results are given and presented graphically and the thermal force was applied. Comparisons are made with the results predicted by (3PHL), (G-N III) and (L-S) in the presence and absence of two-temperature. The boundary plane surface is heated by a non-Gaussian laser beam.
Research Interests:
The model of the equations of generalized magneto-thermoelasticity in an isotropic elastic medium with an initial stress is established. The entire elastic medium is rotated with a uniform angular velocity. The formulation is applied... more
The model of the equations of generalized magneto-thermoelasticity in an isotropic elastic medium with an initial stress is established. The entire elastic medium is rotated with a uniform angular velocity. The formulation is applied under three theories of generalized thermoelasticity: Lord-Schulman with one relaxation time, Green-Lindsay with two relaxation times, as well as the coupled theory. The normal mode analysis is used to obtain the exact expressions for the considered variables. Some particular cases are also discussed in the context of the problem. Numerical results for the considered variables are obtained and illustrated graphically. Comparisons are also made with the results predicted by different theories (CD, L-S, G-L) in the absence and presence of a magnetic field, as well as initial stress.
Research Interests:
In this study, we emphasize on the reflection of elastic waves through a homogenous micro-stretch thermoelastic medium. Material and elastic properties of the medium are considered to be temperature-dependent. The medium selected is... more
In this study, we emphasize on the reflection of elastic waves through a homogenous micro-stretch thermoelastic medium. Material and elastic properties of the medium are considered to be temperature-dependent. The medium selected is rotating about an axis with some fixed angular frequency. Initially, the governing equations are formulated with the help of generalized thermoelastic theory. Heat conduction through the medium is encountered by the model proposed by Green and Naghdi. Governing equations are decomposed into longitudinal and transverse components with the help of Helmholtz decomposition principle. Harmonic representation of wave is used to find the solution to the problem. On the basis of solution, it is concluded that after reflection five quasi-waves propagate through the medium. The quasi term basically represents that the polarization vector of particles is not purely perpendicular or parallel to the propagation vector. The rotational frequency of the medium is respon...
Research Interests:
The present investigation is concerned with the effect of magnetic field on the generalized thermoelastic diffusion with temperature-dependent elastic meduli. The modulus of elasticity is taken as a linear function of reference... more
The present investigation is concerned with the effect of magnetic field on the generalized thermoelastic diffusion with temperature-dependent elastic meduli. The modulus of elasticity is taken as a linear function of reference temperature. The present model is describing the generalization Lord-Shulman theory. The expressions for the temperature, displacement components, the thermal stress distributions and the concentration of diffusion are obtained by using the normal mode method. Numerical results are obtained and illustrated graphically for the problem considered.
In this study we want to decide whether the decay of the solutions of the mixed initial boundary value problem in the context of thermoelasticiy of micropolar bodies with voids is sufficiently fast to guarantee that they vanish after a... more
In this study we want to decide whether the decay of the solutions of the mixed initial boundary value problem in the context of thermoelasticiy of micropolar bodies with voids is sufficiently fast to guarantee that they vanish after a finite time. In fact, we prove that the effect of the micropolar structure in combination with the thermal and porous dissipation can not determine the thermomechanical deformations vanish after a finite time.
Three-phase-lag theory of thermoelasticity is employed to study the deformation of thermo-elastic solid half-space under hydrostatic initial stress, rotation, magnetic field and gravity with two-temperature. The normal mode analysis is... more
Three-phase-lag theory of thermoelasticity is employed to study the deformation of thermo-elastic solid half-space under hydrostatic initial stress, rotation, magnetic field and gravity with two-temperature. The normal mode analysis is used to obtain the analytical expressions of the displacement components, force stress, thermodynamic temperature and conductive temperature. The numerical results are given and presented graphically when mechanical and thermal force is applied. Comparisons are made with the results predicted by the three-phase-lag model, Green-Naghdi III and Lord-Shulman theories.
23 Abstract— The present investigation is concerned with effect of rotation on the wave propagation of generalized thermoelastic diffusion with one relaxation time. The exact expressions for temperature, the displacement components,... more
23 Abstract— The present investigation is concerned with effect of rotation on the wave propagation of generalized thermoelastic diffusion with one relaxation time. The exact expressions for temperature, the displacement components, stress components, chemical potential and concentration are obtained and represented graphically. Numerical computations are carried out and are obtained by using the normal mode analysis. Comparisons are made for different values of relaxation time in the presence and absence of rotation.
Research Interests:
This paper's main objective is to study the two-dimensional deformation of the thermoelastic micro-elongated solid with the effect of initial stress on the model of dual-phase-lag and the theory of Lord and Shulman (L-S). The... more
This paper's main objective is to study the two-dimensional deformation of the thermoelastic micro-elongated solid with the effect of initial stress on the model of dual-phase-lag and the theory of Lord and Shulman (L-S). The interface of the elastic half-space and the thermoelastic micro-elongated half-space is utilized to apply mechanical force. The solution has been made using the normal mode analysis method. For aluminum epoxy, numerical simulations are done and graphically presented to demonstrate the presence and complete absence of initial stress on the various physical quantities.
Research Interests:
This study considers a thermoelastic micro-elongated solid which is being heated by a laser pulse in order to investigate the corresponding impact of thermal loading and magnetic field. A laser beam that is non-Gaussian in nature is used... more
This study considers a thermoelastic micro-elongated solid which is being heated by a laser pulse in order to investigate the corresponding impact of thermal loading and magnetic field. A laser beam that is non-Gaussian in nature is used to heat the surface of the bounded plane with a pulse duration [Formula: see text][Formula: see text]ps. The dual-phase-lag (DPL) model is considered to develop a better understanding of the problem. Also, the problems are solved using normal mode analysis as well as it helps in attaining the appropriate expressions for micro-rotation, components of displacement, the force stress, couple stress, concentration field, the temperature distribution the scalar micro-elongated function, and the micro-stress. The results obtained from the thermal variations are validated by comparing them to Coupled Theory (CT) of thermo-elasticity, Lord and Shulman (L–S) theory, and the DPL model. The graphical illustration of these comparisons has been presented as it he...
Research Interests:
Research Interests:
Research Interests:
In this paper, The aim of this study is to present a reliable combination of the shifted Legendre collocation method to approximate of the problem of free convection boundary-layer flow over a vertical plate as produced by a body force... more
In this paper, The aim of this study is to present a reliable combination of the shifted Legendre collocation method to approximate of the problem of free convection boundary-layer flow over a vertical plate as produced by a body force about a flat plate in the direction of the generating body force. The proposed method is based on replacement of the unknown function by truncated series of well known shifted Legendre expansion of functions. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shift Legendre coefficients. Thus, by solving this system of equations, the shifted Legendre coefficients are obtained. Boundary conditions in an unbounded domain, i.e. boundary condition at infinity, pose a problem in general...
Research Interests:
This paper studies the effect of the magnetic field on a two-dimensional thermoelastic medium based on the Lord–Shulman theory of the body, which has a doubly porous structure. The normal mode analysis is used to obtain the analytic... more
This paper studies the effect of the magnetic field on a two-dimensional thermoelastic medium based on the Lord–Shulman theory of the body, which has a doubly porous structure. The normal mode analysis is used to obtain the analytic expressions of the physical quantities that we consider the solution form in the exponential image. In addition, a homogeneous double porosity thermoelastic half-space with the effect of the Lorentz force has been discussed. Besides, the equations of the generalized double porosity structure thermoelastic material with one relaxation time have been developed. The general solutions of the problem were found, under certain conditions. Moreover, the numerical results are discussed graphically in comparisons with the presence and absence of the magnetic field for copper material.