1. bookAHEAD OF PRINT
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
access type Open Access

Human Body Movement Coupling Model in Physical Education Class in the Educational Mathematical Equation of Reasonable Exercise Course

Published Online: 27 Jul 2021
Page range: -
Received: 12 Apr 2021
Accepted: 07 Jul 2021
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

In physical education class, the movement of the human body requires multiple joints to cooperate, and a multi-link system coupling is presented. In the teaching of physical education curriculum, the impact of the force received by students jumping up and down shows the characteristics of the non-linear system of physics and mathematics. Aiming at the movement process of jumping up and down, we established a joint mathematical equation model of the motion state of the human lower limb joints. We use a non-linear system to solve the mathematical model of the joint force coupling problem of the human body jumping up and down.

Keywords

MSC 2010

Introduction

Analyzing human movement from biomechanical characteristics is a multi-link chain system characteristic. Different joints play different roles in this multi-link limb chain. At present, there is not much research on the multi-joint motion chain of the lower limbs, and the mechanism of motion control is not precise. The kinematic characteristics of the knee joint when the human body falls from a height are important factors that affect the ground reaction force. The impact at the moment of landing is a complex physical process, and the human body is viscoelastic [1]. Therefore, the human body has a series of dynamic responses to the impact at the moment of landing. This article attempts to use the dominant joint decomposition theory to establish a collaborative mathematical model of the lower limb system at the moment of landing. This article hopes that the conclusions drawn in the thesis will provide a theoretical basis for analyzing the complex motion mechanism of lower limbs.

Mathematical model of the foot and ankle joint at the moment of human landing

The thesis defines the moment when the foot and ankle joints are in two states during the moment when the human body lands from a height to land (using a single foot as the research object). The first state is when the toe touches the ground, and the second state is when the foot is flat [2]. The moment the toe touches the environment comes from the muscle Achilles tendon force. For the ankle system, it is the longitudinal internal force F, and on the vertical plane F1 + FG = mg + FG and F1 = mg. {F1=mgF1l1sinα1=Fl1cosα1FGl2sinα2=Fl2cosα2\begin{equation}\left\{\begin{array}{l}F_{1}=m g \\F_{1} l_{1} \sin \alpha_{1}=F l_{1} \cos \alpha_{1} \\F_{G} l_{2} \sin \alpha_{2}=F l_{2} \cos \alpha_{2}\end{array}\right.\end{equation}

We put together formula (1) to get a mathematical model of the moment the toe touches the ground {F1=mgF=F1tanα1FG=F/tanα2\begin{equation}\left\{\begin{array}{l}F_{1}=m g \\F=F_{1} \tan \alpha_{1} \\F_{G}=F / \tan \alpha_{2}\end{array}\right.\end{equation}

Fig. 1

Schematic diagram of the human body landing from the toe touching the ground to the foot lying flat.

For the second state, the foot is always flat, assuming that the human body maintains a balanced and static condition F1 + F2 = mg. Decomposing from the moment of the ankle joint we get: {F1l1sinα1=Fl1cosα1F2l2sinα2=Fl2cosα2\begin{equation}\left\{\begin{array}{l}F_{1} l_{1} \sin \alpha_{1}=F l_{1} \cos \alpha_{1} \\F_{2} l_{2} \sin \alpha_{2}=F l_{2} \cos \alpha_{2}\end{array}\right.\end{equation}

Organize formula (2) to get {F1=F/tanα1F2=F/tanα2\begin{equation}\left\{\begin{array}{l}F_{1}=F / \tan \alpha_{1} \\F_{2}=F / \tan \alpha_{2}\end{array}\right.\end{equation}

From this we get the mathematical model of the human foot and ankle joint {F1+F2=mgF1=F/tanα1F2=F/tanα2\begin{equation}\left\{\begin{array}{l}F_{1}+F_{2}=m g \\F_{1}=F / \tan \alpha_{1} \\F_{2}=F / \tan \alpha_{2}\end{array}\right.\end{equation}

From model (2) and formula (5), the joint mathematical model of the ankle joint of the human body at the moment of landing impact is obtained. F1 is the ground reaction force at the forefoot (forefoot). F2 is the ground reaction force at the back of the foot (heel). m is the weight of the human body. g is the acceleration due to gravity. F is the resultant force of the longitudinal structure. l1 The distance from the sole to the center of the ankle joint. l1 The length from the sole of the back of the foot to the center of the ankle joint. α1 is the angle between the line between the sole of the forefoot and the center of the ankle common and the vertical (y) direction. α2 is the angle between the line from the sole of the back of the foot to the center of the ankle joint and the straight movement (y).

From this model, the longitudinal structural force of the foot can be calculated [3]. For the soles of the same size, the lower the arch (flat feet), the greater the angle between α1 and α2. The greater the longitudinal structural stress acting on the sole that maintains the integrity of the turn. This causes excessive pressure on the connective tissue of the plantar, and the clinical response is to cause plantar fascia disease. The moment the toe touches the ground, the force on the longitudinal arch structure increases significantly. At the same time, a significant internal pressure occurs at the Achilles tendon inside the ankle system. At this time, people with plantar fascia pain will adapt to the pathological condition.

Mathematical model of knee and hip system at the moment of impact of human landing

According to the leading joint decomposition theory, we disassemble the common knee axis. We assume that both the hip joint and the knee joint rotate shaft structures around the center [4]. The direction of the axis of rotation is perpendicular to the forward direction. The mathematical model of the knee joint is as follows: {FCcosθC=mCaCcosφCFCsinθC=mCaCsinφC+mCg\begin{equation}\left\{\begin{array}{l}F_{C} \cos \theta_{C}=m_{C} a_{C} \cos \varphi_{C} \\F_{C} \sin \theta_{C}=m_{C} a_{C} \sin \varphi_{C}+m_{C} g\end{array}\right.\end{equation}

Fig. 2

Schematic diagram of the force on the lower limb system at the moment of landing.

The mathematical model of the hip joint obtained by inverse dynamics is as follows: {FCcosθC+FDcosθD=mDαDcosφDFCsinθC+FDsinθD=mDαDsinφD+mDg\begin{equation}\left\{\begin{array}{l}-F_{C} \cos \theta_{C}+F_{D} \cos \theta_{D}=m_{D} \alpha_{D} \cos \varphi_{D} \\-F_{C} \sin \theta_{C}+F_{D} \sin \theta_{D}=m_{D} \alpha_{D} \sin \varphi_{D}+m_{D} g\end{array}\right.\end{equation}

We can obtain the combined mathematical model of the knee joint and hip joint torque from the angular movement of the calf around the knee joint {MCmCglCcosβC=(IC+mClC2)αCMDMCmDglDcosβD=(ID+mDlD2)αD\begin{equation}\left\{\begin{array}{l}M_{C}-m_{C} g l_{C} \cos \beta_{C}=\left(I_{C}+m_{C} l_{C}^{2}\right) \alpha_{C} \\M_{D}-M_{C}-m_{D} g l_{D} \cos \beta_{D}=\left(I_{D}+m_{D} l_{D}^{2}\right) \alpha_{D}\end{array}\right.\end{equation}

FC, FD is the resultant force on the knee joint and hip joint, respectively. θC, θD is the angle between the knee joint and hip joint force and the horizontal direction (x), respectively. MC, MD is the resultant moment on the knee joint and hip joint, respectively. mC, mD is the mass of calf and thigh, respectively. aC, aD is the acceleration on the mass center of the calf and thigh, respectively. ϕC, ϕD is the angle between the force generated by the mass center of the calf and the thigh and the horizontal direction (x). lC is the distance from the center of the knee joint to the center of mass of the calf. lD is the distance from the center of the hip joint to the center of mass of the thigh. βC is the angle between the calf and the horizontal direction (x), βD and is the angle between the thigh and the horizontal direction (x). αC, αD is the angular acceleration of the knee joint and hip joint, respectively. IC, ID is the moment of inertia of the calf and thigh, respectively. g is the acceleration of gravity. The combined torque represents the movement of the joint flexor muscles [5]. Since the human body is a complex structure, muscle contraction causes it to produce a resultant moment.

Analysis of the mathematical model of the human body's lower limb system motion coupling problem based on the Hamilton system

We assume f˙=θf\dot{f} = \partial_\theta f and f = x f, considering the dynamic coupling problem, that is, the Lagrange function is: L(ε)=F(Rr)2r2ε2+G2[(ε)2+(ε¨)2+2μβε¨+2(1μ)(ε˙')2]+Nx2(ε)2Nxθε'ε˙\begin{equation}L(\varepsilon)=\frac{F(R-r)}{2 r^{2}} \varepsilon^{2}+\frac{G}{2}\left[\left(\varepsilon^{\prime \prime}\right)^{2}+(\ddot{\varepsilon})^{2}+2 \mu \beta \ddot{\varepsilon}+2(1-\mu)\left(\dot{\varepsilon}^{\prime}\right)^{2}\right]+\frac{N_{x}}{2}(\varepsilon)^{2}-N_{x \theta} \varepsilon^{\prime} \dot{\varepsilon}\end{equation}

We use the space coordinate to simulate the time coordinate t, according to the Hamilton variational principle: δ{F(Rr)2r2ε2+G2[(ε)2+(ε¨)2+2μεω¨ε¨+2(1μ)(ε˙')2+Nx2(ε')2Nxθε'ε}rdθdx=0\begin{equation}\delta \iint\left\{\frac{F(R-r)}{2 r^{2}} \varepsilon^{2}+\frac{G}{2}\left[\left(\varepsilon^{\prime \prime}\right)^{2}+(\ddot{\varepsilon})^{2}+2 \mu \varepsilon^{\prime \prime} \ddot{\omega} \ddot{\varepsilon}+2(1-\mu)\left(\dot{\varepsilon}^{\prime}\right)^{2}+\frac{N_{x}}{2}\left(\varepsilon^{\prime}\right)^{2}-N_{x \theta} \varepsilon^{\prime} \varepsilon\right\} r d \theta d x=0\right.\end{equation}

We double-integrate the formula (2) to get: δ{F(Rr)2r2ε2+G2[(ε)2+(ε¨)2+2(ε)2]+Nx2(ε')2Nxθε'ε˙}rdθdx=0\begin{equation}\delta \iint\left\{\frac{F(R-r)}{2 r^{2}} \varepsilon^{2}+\frac{G}{2}\left[\left(\varepsilon^{\prime \prime}\right)^{2}+(\ddot{\varepsilon})^{2}+2\left(\varepsilon^{\prime \prime}\right)^{2}\right]+\frac{N_{x}}{2}\left(\varepsilon^{\prime}\right)^{2}-N_{x \theta} \varepsilon^{\prime} \dot{\varepsilon}\right\} r d \theta d x=0\end{equation}

We set the initial variable q={ε,ε˙}q = \{\varepsilon,-\dot{\varepsilon}\} and the dual variable p = {p1, p2}. p1=Qθ=D(ε˙+ε)p_{1}=Q_{\theta}=-D\left(\dot{\varepsilon}^{\prime \prime}+\varepsilon\right) , p2=(Mx+Mθ)/[1+(Rr)]=D(ε+ε¨)p_{2}=\left(M_{x}+M_{\theta}\right) /[1+(R-r)]=-D\left(\varepsilon^{\prime \prime}+\ddot{\varepsilon}\right) , p1 is the equivalent shear force in the θ direction and p2 is the comparable torsional moment in the θ order [6]. The Hamilton function that we use the initial variable and the dual variable to express is: M(q,p)=pTqL(q,p)=F(Rr)2r2ε2+12Gp22+p1ε+p2εNx2(ε')2+Nxθε'ε\begin{equation}M(q, p)=p^{T} q-L(q, p)=-\frac{F(R-r)}{2 r^{2}} \varepsilon^{2}+\frac{1}{2 G} p_{2}^{2}+p_{1} \varepsilon+p_{2} \varepsilon^{\prime \prime}-\frac{N_{x}}{2}\left(\varepsilon^{\prime}\right)^{2}+N_{x \theta} \varepsilon^{\prime} \varepsilon\end{equation}

The Hammon equation is described as follows: {ε˙ε˙p˙1p˙2}={δM/δP1δM/δP1δM/δεδM/δε˙}=[0100x2ε001/DF(Rr)r2+Nxx2εNxθxε0x2εNxθxη010]{εp2p1p2}\begin{equation}\left\{\begin{array}{l}\dot{\varepsilon} \\-\dot{\varepsilon} \\\dot{p}_{1} \\\dot{p}_{2}\end{array}\right\}=\left\{\begin{array}{l}\delta M / \delta P_{1} \\\delta M / \delta P_{1} \\-\delta M / \delta \varepsilon \\\delta M / \delta \dot{\varepsilon}\end{array}\right\}=\left[\begin{array}{cccc}0 & -1 & 0 & 0 \\\partial_{x}^{2} \varepsilon & 0 & 0 & 1 / D \\\frac{F(R-r)}{r^{2}}+N_{x} \partial_{x}^{2} \varepsilon & -N_{x \theta} \partial_{x} \varepsilon & 0 & -\partial_{x}^{2} \varepsilon \\N_{x \theta} \partial_{x} \eta & 0 & 1 & 0\end{array}\right]\left\{\begin{array}{l}\varepsilon \\-p_{2} \\p_{1} \\p_{2}\end{array}\right\}\end{equation}

We abbreviate formula (13) as: ψ˙=Hψ\begin{equation}\dot{\psi} = H\psi\end{equation}

The entire state vector: ψ={qT,pT}={ε,ε˙,p1,p2}\begin{equation}\psi=\left\{q^{T}, p^{T}\right\}=\left\{\varepsilon,-\dot{\varepsilon}, p_{1}, p_{2}\right\}\end{equation}

The operator matrix is: M=[0100x2ε001/GE(Rr)r2+Nxx2εNxθxε0x2εNxθxε010]\begin{equation}M=\left[\begin{array}{cccc}0 & -1 & 0 & 0 \\\partial_{x}^{2} \varepsilon & 0 & 0 & 1 / G \\\frac{E(R-r)}{r^{2}}+N_{x} \partial_{x}^{2} \varepsilon & -N_{x \theta} \partial_{x} \varepsilon & 0 & -\partial_{x}^{2} \varepsilon \\N_{x \theta} \partial_{x} \varepsilon & 0 & 1 & 0\end{array}\right]\end{equation}

The identity matrix is expressed as J=[0I2I20]\begin{equation}J = \left[\begin{array}{cc}0 & I_2 \\-I_2 & 0 \end{array}\right]\end{equation}

We have ψ1,ψ2=0xeψ1Jψ2dx\begin{equation}\left\langle\psi_{1}, \psi_{2}\right\rangle=\int_{0}^{x_{e}} \psi_{1} J \psi_{2} d x\end{equation}

ψ = {ɛ, − ɛ, p1, p2} forms a symplectic space vector. In the linear Hamilton system, we use the generalized variable separation method to make ψ(x,θ)=ϕ(x)eλθ\begin{equation}\psi(x, \theta)=\phi(x) e^{\lambda \theta}\end{equation}

φ(x) is the intrinsic solution, which is a function of x. The Eigen equation is expressed as follows: Hϕ=λϕ\begin{equation}H\phi = \lambda\phi\end{equation}ϕ(x)=ϕ(x)e2λπ\begin{equation}\phi(x)=\phi(x) e^{2 \lambda \pi}\end{equation}

Among them, when λ = in(n = 0,±1,…),n = 0 is the eigenvalue λ = 0, the formula (12) becomes = 0. At this time, the characteristic equation is η42n2r2η2+n4r4+E(Rr)Dr3=0\begin{equation}\eta^{4}-\frac{2 n^{2}}{r^{2}} \eta^{2}+\frac{n^{4}}{r^{4}}+\frac{E(R-r)}{D r^{3}}=0\end{equation}

φ(x) has nothing to do with N. When n ≠ 0, the eigenvalue λn ≠ 0,φn(x) is a non-zero eigensolution. If λi + λj ≠ 0, then ϕi,ϕj=0xeϕiTJϕjdx=0\begin{equation}\left\langle\phi_{i}, \phi_{j}\right\rangle=\int_{0}^{x_{e}} \phi_{i}^{T} J \phi_{j} d x=0\end{equation}

Solutions φi and φj have a symplectic orthogonal relationship. ϕi,ϕj=0xeϕiTJϕjdx0\begin{equation}\left\langle\phi_{i}, \phi_{j}\right\rangle=\int_{0}^{x_{e}} \phi_{i}^{T} J \phi_{j} d x \neq 0\end{equation}

The relationship between φj and φj is symplectic conjugate [7]. The vector space is complete, and any eigenstate vector can be represented by a combination of function eigenvectors. ψ(x,θ)=n=0[anϕn(x)einθ+bnϕn(x)einθ]\begin{equation}\psi(x, \theta)=\sum_{n=0}^{\infty}\left[a_{n} \phi_{n}(x) e^{i n \theta}+b_{n} \phi-n(x) e^{-i n \theta}\right]\end{equation}

We can get this by solving the Eigen equation: ϕn(x)=k=14CkxknAkn(x)=k=14Ckxkneηkxk=14Ck{1in/rD(in3/r3+ηkn2/r2)D(n2/r2ηk2)}eηkx\begin{equation}\phi_{n}(x)=\sum_{k=1}^{4} C_{k} x_{k}^{n} A_{k}^{n}(x)=\sum_{k=1}^{4} C_{k} x_{k}^{n} e^{\eta k x} \sum_{k=1}^{4} C_{k}\left\{\begin{array}{l}1 \\-i n / r \\D\left(i n^{3} / r^{3}+\eta k n^{2} / r^{2}\right) \\D\left(n^{2} / r^{2}-\eta_{k}^{2}\right)\end{array}\right\} e^{\eta k x}\end{equation}

Ck,(k = 1,2,3,4) is the undetermined Eigen coefficient [8]. The eigenvalue equation of symplectic space is shown below. ηk,(k = 1,2,3,4) is the solution of the Eigen-characteristic equation in symplectic space. η4+(NxDr22n2r2)η2+2niNxyDr2η+n4r4+E(Rr)Dr3=0\begin{equation}\eta^{4}+\left(\frac{N_{x}}{D r^{2}}-\frac{2 n^{2}}{r^{2}}\right) \eta^{2}+\frac{2 n i N_{x y}}{D r^{2}} \eta+\frac{n^{4}}{r^{4}}+\frac{E(R-r)}{D r^{3}}=0\end{equation}

Simulation of calculation results of lower limb system at the moment of human landing impact

We simulate the action of the human body jumping from a height of 66 cm. The load is the mechanical parameter that occurs in the dynamic experiment [9]. When the subject is landing from a height, the angle between the extension line of the calf.

Fig. 3

Schematic diagram of the simulation of a human body landing from the toe touching the ground to the foot lying flat.

From the simulation results, it can be seen that the knee joint has an excellent cushioning effect on the vertical ground reaction force, and the angular velocity of the hip joint has an advantage in cushioning the horizontal backward reaction force.

We set the unit properties of the material of the parts and the contact between the components. The model setting boundary conditions and loads have been developed. During the simulation, a rotational load was applied to the femur during buckling [10]. We simulate the internal and external rotation of the femur, and the angle of internal and external rotation is gradually increased from 5° to 30°, as shown in Figure 4. We set the measurement parameters and then solve them. The result is shown in Figure 5 [11].

Fig. 4

Schematic diagram of the momentary load of the human lower limbs landing.

Fig. 5

The stress cloud surrounding the lower limbs of the human body at the moment of landing.

The landing method that reduces the maximum vertical reaction force has a greater degree of joint flexion than the proper landing method [12]. Therefore, the human body delays the cushioning of the landing process by increasing the degree of flexion of the joints and reduces the reaction force of the ground under the state of falling motion-fixed.

The skeletal muscle system of the lower limb system of the human body immediately reaches a large load during the landing. Therefore, the method used to control the reaction force must be activated before touching the ground. Thus, the muscles of the lower limbs of the human body enter the contraction state before the end of the link chain touches the ground, and the joint chain should be in a flexion state before touching the ground [13].

Conclusion

According to the decomposition theory of dominant joints, a joint mathematical model of the lower limb system of the human body is established at the moment of landing. Based on the Hamiltonian system, the mathematical model of the coupling problem of the lower limbs of the human body is analyzed. From the simulation results, it can be known that since the skeletal muscle system reaches a large load immediately when it touches the ground, the method used to control the reaction force must be activated before the ground touches.

Fig. 1

Schematic diagram of the human body landing from the toe touching the ground to the foot lying flat.
Schematic diagram of the human body landing from the toe touching the ground to the foot lying flat.

Fig. 2

Schematic diagram of the force on the lower limb system at the moment of landing.
Schematic diagram of the force on the lower limb system at the moment of landing.

Fig. 3

Schematic diagram of the simulation of a human body landing from the toe touching the ground to the foot lying flat.
Schematic diagram of the simulation of a human body landing from the toe touching the ground to the foot lying flat.

Fig. 4

Schematic diagram of the momentary load of the human lower limbs landing.
Schematic diagram of the momentary load of the human lower limbs landing.

Fig. 5

The stress cloud surrounding the lower limbs of the human body at the moment of landing.
The stress cloud surrounding the lower limbs of the human body at the moment of landing.

Trenchev, I., Mavrevski, R., Traykov, M., & Zajmi-Rugova, I. A mathematical model of movement in virtual reality through thoughts. International Journal of Electrical and Computer Engineering (IJECE)., 2020.10(6): 6592–6597TrenchevI.MavrevskiR.TraykovM.Zajmi-RugovaI.A mathematical model of movement in virtual reality through thoughtsInternational Journal of Electrical and Computer Engineering (IJECE)202010665926597Search in Google Scholar

Pastor, D., Campayo-Piernas, M., Pastor, J. T., & Reina, R. A mathematical model for decision-making in the classification of para-footballers with different severity of coordination impairments. Journal of sports sciences., 2019. 37(12): 1403–1410PastorD.Campayo-PiernasM.PastorJ. T.ReinaR.A mathematical model for decision-making in the classification of para-footballers with different severity of coordination impairmentsJournal of sports sciences2019371214031410Search in Google Scholar

König, T., Kux, H. J., & Mendes, R. M. Shalstab mathematical model and WorldView-2 satellite images to identification of landslide-susceptible areas. Natural Hazards., 2019. 97(3): 1127–1149.KönigT.KuxH. J.MendesR. M.Shalstab mathematical model and WorldView-2 satellite images to identification of landslide-susceptible areasNatural Hazards201997311271149Search in Google Scholar

Bapat, G. M., & Sujatha, S. A Two-Dimensional Mathematical Model to Simulate the Effects of Knee Center Mis-alignment in Lower-Limb Orthoses. JPO: Journal of Prosthetics and Orthotics., 2021. 33(1): 34–45BapatG. M.SujathaS.A Two-Dimensional Mathematical Model to Simulate the Effects of Knee Center Mis-alignment in Lower-Limb OrthosesJPO: Journal of Prosthetics and Orthotics20213313445Search in Google Scholar

Arguello, J. R., Laurent, S., & Clark, A. G. Demographic history of the human commensal Drosophila melanogaster. Genome biology and evolution., 2019. 11(3): 844–854ArguelloJ. R.LaurentS.ClarkA. G.Demographic history of the human commensal Drosophila melanogasterGenome biology and evolution2019113844854Search in Google Scholar

Paterson, C., Clevers, H., & Bozic, I. Mathematical model of colorectal cancer initiation. Proceedings of the National Academy of Sciences., 2020. 117(34): 20681–20688PatersonC.CleversH.BozicI.Mathematical model of colorectal cancer initiationProceedings of the National Academy of Sciences2020117342068120688Search in Google Scholar

Hatanaka, S., & Ishii, N. Proposal and validation of mathematical model for resistance training. The Journal of Physical Fitness and Sports Medicine., 2021. 10(2): 109–118HatanakaS.IshiiN.Proposal and validation of mathematical model for resistance trainingThe Journal of Physical Fitness and Sports Medicine2021102109118Search in Google Scholar

Fakhrzad, M. B., & Goodarzian, F. A new multi-objective mathematical model for a Citrus supply chain network design: Metaheuristic algorithms. Journal of Optimization in Industrial Engineering., 2021. 14(2): 127–144FakhrzadM. B.GoodarzianF.A new multi-objective mathematical model for a Citrus supply chain network design: Metaheuristic algorithmsJournal of Optimization in Industrial Engineering2021142127144Search in Google Scholar

Baskonus, H. M., Bulut, H., & Sulaiman, T. A. New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion method. Applied Mathematics and Nonlinear Sciences, 2019. 4(1): 141–150.BaskonusH. M.BulutH.SulaimanT. A.New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion methodApplied Mathematics and Nonlinear Sciences201941141150Search in Google Scholar

Aidara, S. Anticipated backward doubly stochastic differential equations with non-Liphschitz coefficients. Applied Mathematics and Nonlinear Sciences, 2019. 4(1): 9–20.AidaraS.Anticipated backward doubly stochastic differential equations with non-Liphschitz coefficientsApplied Mathematics and Nonlinear Sciences201941920Search in Google Scholar

Ali, A., Hussain, M., Ghaffar, A., Ali, Z., Nisar, K. S., Alharthi, M. R., & Jamshed, W. Numerical simulations and analysis for mathematical model of avascular tumor growth using Gompertz growth rate function. Alexandria Engineering Journal., 2021. 60(4): 3731–3740AliA.HussainM.GhaffarA.AliZ.NisarK. S.AlharthiM. R.JamshedW.Numerical simulations and analysis for mathematical model of avascular tumor growth using Gompertz growth rate functionAlexandria Engineering Journal202160437313740Search in Google Scholar

Valnes, L. M., Mitusch, S. K., Ringstad, G., Eide, P. K., Funke, S. W., & Mardal, K. A. Apparent diffusion coefficient estimates based on 24 hours tracer movement support glymphatic transport in human cerebral cortex. Scientific reports., 2020. 10(1): 1–12ValnesL. M.MituschS. K.RingstadG.EideP. K.FunkeS. W.MardalK. A.Apparent diffusion coefficient estimates based on 24 hours tracer movement support glymphatic transport in human cerebral cortexScientific reports2020101112Search in Google Scholar

Resmawan, R., & Yahya, L. Sensitivity analysis of mathematical model of coronavirus disease (COVID-19) transmission. Cauchy., 2020. 6(2): 91–99ResmawanR.YahyaL.Sensitivity analysis of mathematical model of coronavirus disease (COVID-19) transmissionCauchy2020629199Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo