Mathematics Department, UIN Maulana Malik Ibrahim Malang, Malang, Indonesian;
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1 P r o c e e d i n g I n t e r n a t i o n a l C o n f e r e n c e, 0 3, *, * * - * * The 4 th Green Technology Faculty of Science and Technology Islamic of University State Maulana Malik Ibrahim Malang SOLUTION OF VAN DER POL EQUATION USING ADAMS BASHFORTH MOULTON FOURTH ORDER METHODS NurAzizah, Ari Kusumastuti, S.Si, M.Pd Mathematics Department, UIN Maulana Malik Ibrahim Malang, Malang, Indonesian; Mathematics Department, UIN Maulana Malik Ibrahim Malang, Malang, Indonesian. n4z4_azizah6@yahoo.com ABSTRACT Problems involving mathematical models, especially the form of differential equations often arise in the application. For example, the form of ordinary differential equations the process Van der Pol derived from RLC circuit problem. Van der Pol equation obtained from the research are studied Balthazar Van der Pol in 90 for the same type the RLC circuit, but a passive resistor from Ohm's law is replaced by an active element formed from a closed triode tubes (semiconductor). This equation is a form of non-linear differential equations are difficult to solve analytically, so the solution can be done numerically, of whom can use method of fourth order Adams Bashforth Moulton (ABM). In this study, the completion of the Van der Pol equation using ABM fourth order methods where each of the three initial values and is obtained from the Runge Kutta (RK) method. Solution is initially smaller gradually increased in amplitude and the solution is also smaller increases gradually so that each oscillation solution reaches a certain limit. Further analysis of the dynamic behavior of the Van der Pol equation shows that the Van der Pol equation around the fixed point (0,0) is an unstable spiral point. All trajectories move towards a single periodic orbit. Key Words: Adams Bashforth Moulton (ABM) Fourth Order Methods, Van der Pol Equation. 344
2 Nur Azizah et al. * (03) **-** INTRODUCTION Mathematical model in the form of differential equations frequently arise in the application, for example, Van der Pol equation. Completion of the equation can not be solved analytically as the Van der Pol equation, then the solution can be done numerically. One of the numerical methods that can be used is ABM fourth order methods. It, first used the formula predictor to predict a value of corrector formula is then used to correct the value of better (Djojodihardjo, 000). Then the behavior of solutions of Van der Pol equation is done by analyzing near fixed point and trajectory stability Van der Pol equation in phase plane. BASIC THEORY. Van der Pol Equation The research studied Balthazar Van der Pol in 90 for the same type the RLC circuit, but a passive resistor from Ohm's law is replaced by an active element formed from a closed tube triode (semiconductor) as shown Fig. (Tsatsos, 006). Van der Pol equation is shown as follows: where μ is a parameter damped. So the Van der Pol equation system is obtained and (..3) external + E(t) voltage - semikonduktor R I(t) C induktor capacitor Figure. Sircuit RLC of semiconductor. ABM Fourth Order Methods ABM method fourth order formula for second order differential equations are as follows: Predictor: L (..) and (Bronson and Costa, 007). 3. Error of ABM Fourth Order Methods The error predictor of Adams Bashforth (AB) and corrector of Adams Moulton (AM) in the order of in the differential equation, that (.3.) (.3.) If is the exact value of at, then an error estimate from equation (.3.) and (.3.): (.3.3) (.3.4) Generally, but, if it considers that the relevant interval remains close, then after reducing equation (.3.3) from (.3.4) obtained the following estimates for is. If this is substituted into equation (.3.), so (.3.5) (Conte and Boor, 993). 4. Dynamic Analysis in Autonomous 4. Autonomous and Nonautonomous Let system of differential equations and (.4.) If and do not depend explicitly on, it is called autonomous systems. Conversely, if and depend explicitly on, it is called nonautonomous system (Hariyanto, et al., 99). 4. Fixed Point of Autonomous Fixed points of system (.4.) is, such that and (Waluya, 006). Corrector: 345
3 Nur Azizah et al. * (03) **-** 4.3 Eigen Value, Eigen Vector, and Combination Linear of Solution Let system of linear differential equations (.4.),, and. is a eigenvalues for matrix if only if can t inversed (Conte dan Boor, 993). There (.4.3) For each eigenvalue, there is a corresponding nonzero solution, called an eigenvector, so obtained (.4.4) where is the identity matrix and 0 is zero vector. For each eigenvalue-eigenvector pair there is a corresponding vector solution of Equation (.4.). A general solution of matrix is combination linear of (.4.5) where can obtained given initial condition at Equation (.4.) (Boyce dan DiPrima, 999). Assume that is an matrix solution of Equation (.4.). The determinant is called the Wronskian of the linear system. The solutions are called independent, provided that the corresponding matrix solution has (Robinson, 004). 4.4 Analysis Phase Plane of Autonomous Let autonomous system is and (.4.6). Characteristic equation of Equation (.4.6) is (.4.7) Stability of fixed point for Equation (.4.6) depended value and of Equation (.4.7). This stability show in table as below: Table. Stability of Equation (.4.6) Eigenvalues Type of Critical Point Stability Node Node Asymptotically stable Saddle point Proper or improper node Proper or improper Asymptotically stable 346 node Spiral point Center Asymptotically stable Stable (Boyce dan DiPrima, 00). 4.5 Linearization Let autonomous systems (.4.) and is nonlinear, using the Taylor expansion near at the time, so obtained the linearized system at a fixed point is given by (.4.8) where all the partial derivatives in the matrix are evaluated at (Robinson, 004). 5. Parameter Van der Pol Equation Van der Pol equation has a relevant interest, particularly in the extreme cases when the parameter is either small or very large, which are associated typical asymptotic behaviors of self-oscillating systems it describes. When is very large, in the limit tending to infinity, one obtains relaxation oscillations, i.e., strongly non linear oscillations exhibiting sharp periodic jumps. Typical examples of such systems are nearly sinusoidal electronic oscillators and multivibrators (Buonomo,998). DISCUSSION. Solution of Van der Pol Equation Using ABM Fourth Order Methods In Equation (..3) show that and given,,, and take the value of and. So many iterations that is iterations. Further find for value using Equation (..3) and (.3.5) and,, RK fourth order methods for starting iteration so obtained as table below:
4 y(t) x(t), y(t) Nur Azizah et al. * (03) **-** Tabel 3. Solution of Van der Pol Equation Using ABM Fourth Order Methods Plot. 3 Solusi Persamaan Van der Pol, = x y. Eigenvalues in the form and. This behavior is called a spiral point and for, then the stability of the fixed point is unstable. Then eigenvalues, obtained eigenvectors means that and solution is. This, and the general waktu (t) Figure 3. Plot Value and versus Solusi Persamaan Van der Pol, = x(t) Figure 4. Plot Value versus. Dynamic Analysis at near Fixed Point Van der Pol Equation A Van der Pol system (..3), so Equation (..3) to be (3..) and obtained a fixed point (0,0), it is so the original point. For known behavior of solution of nonlinear Equation (3..) then used approach linear system. Then using Equation (.4.6) obtained the linearized system at a fixed point (0,0) is given by (3..) where and. Next look for the eigenvalues and eigenvectors of equation (.4.4), and the eigenvalues are sin3 3 cos3 sin3 (3..3) because and so and, and obtained value and. Then substituted into Equation (3..3) and obtained the particular solution for Equation (3..) is Then Wronskian of System (3..3) at (3..4) is. So solutions of system (3..3) is independent. Further system (3..) obtained as shown figure (3) and (4). As for the Van der Pol equation (3..) obtained as shown figure (5) and (6). Figure 5. Phase portrait for Equation (3..) spiral point 347
5 Nur Azizah et al. * (03) **-** Figure 6. Plot of and versus for Equation (3..) initial condition (,0) Figure 7. Phase portrait for Equation (3..) Figure 8. Plot and versus for Equation (3..) initial condition (,0) 3. Interpretation Solutions of Equation (..3) by using ABM fourth order methods, produced images (3.) which is similar to the figure (3.6). Figure (3.3) shows that the trajectory moves in a clockwise direction. Behavior of this solution is called an unstable spiral. In the figure (3.4) shows that the value of and are unstable due to infinite. Amplitude of the solutions and increased sharply at an interval of 40 to 50 and so did the period. Figure (3.5) show that all trajectory goes to a unique periodic orbit. Trajectory that moves from a fixed point to the periodic orbit similar moves the movement trajectory on Van der Pol equations are linearized. Solution and is initially 348 smaller in amplitude gradually increased so that each oscillation solution reaches a certain limit. In addition, it was found that the graph and respectively have the periods in the interval = [0,50]. While the analysis of the dynamic behavior of the Van der Pol equation solutions around the fixed point (0,0) is an unstable spiral point as shown in the figure (5). CONCLUSION Based on the research conducted, it can be concluded that the settlement of Van der Pol equation using ABM method fourth order, it was found that at the time, and the value of every step to error for and are respectively and Further analysis of the dynamic behavior of the Van der Pol equation shows that the Van der Pol equation around the fixed point (0,0) is an unstable spiral point. All trajectories move towards a single periodic orbit. REFERENCES Boyce, W.E. & DiPrima, R.C ODE Architect Companion. New York: John Willey & Sons, Inc. Boyce, W.E. &DiPrima, R.C Elementary Differential Equation and Boundary Value Problems Seventh Edition. New York: John Willey & Sons, Inc. Bronson, R. & Costa, G.B Schaum s Outlines: Persamaan Diferensial Edisi Ketiga. Jakarta: Erlangga. Buonomo, A The Periodic Solution of Van der Pol s Equation. Siam J. Appl. Math,Vol. 59, No., pp Conte, S.D. & Boor, C.d Dasar-dasar Analisis Numerik Suatu Pendekatan Algoritma Edisi Ketiga. Jakarta: Erlangga. Djojodihardjo, H Metode Numerik. Jakarta: PT Gramedia Pustaka Utama. Hariyanto, Soehardjo, Sumarno, dan Suharmadi. 99. Persamaan Diferensial Biasa Modul -9. Cetakan ke-.jakarta: Universitas Terbuka. Robinson, R.C An Introduction To Dynamical Continuous and Discrete. New Jersey: Pearson Education Inc. Tsatsos, M Theoretical and Numerical Study of the Van der Pol Equation. Disertasi Tidak Diterbitkan. Thessaloniki: Aristotle University. Waluya, S.B Persamaan Diferensial. Yogyakarta: GrahaIlmu.
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