An Interpolation Method That Minimizes an Energy Integral of Fractional Order

Gunawan, Hendra; Pranolo, F.; Rusyaman, Endang

doi:10.1007/978-3-540-87827-8_12

Cited by 5 publications

(6 citation statements)

References 8 publications

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“…Besides that, M. Hoffer and H. Pottman designed energy-minimizing rigid body motions in the presence of obstacles [9]. Also, H. Gunawan et al developed a fractional order and correlated it with material elasticity [10]. They found that the order of fractional differential equation can be associated with elasticity.…”

Section: Meanwhile Adam Loverro Used the Fractional Derivative In Phmentioning

confidence: 99%

Fractional Differential Equation as a Models of Newton Fluids for Stress and Strain Problems

Rusyaman¹,

Chaerani²,

Parmikanti³

2018

Proceedings of the 1st Annual International Conference on Mathematics, Science, and Education (ICoMSE 2017)

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An ordinary differential equation is a branch of mathematics that is always interesting to be learned and developed due to its numerous variations in both the theory and its application. In general, the most discussed differential equation is the ordinary differential equation that has the natural number as its order. However, currently, the order of a differential equation which has been developed into a fractional-order (i.e. rational numbers) is also very interesting to be studied. This paper presents a study of ordinary differential equations that has fractional order, in which the left side contains two derivative functions with fractional order while the right side contains polynomial function n degree. Specifically, this paper presents the general form of equations, methods of finding the solution in three cases, and its application. Solutions are proposed using the Laplace transformation and its inverse and are expressed in the form of Mittag-Lefler function. Its graph is also later described using Matlab. Results of this research are expressed by three functions of three different theoretical cases and a solution to an application problem. Additionally, the study has also shown that the convergence of a number sequence of fractional differential equation order is positively related to the convergence of solution function sequence. There are many applications of fractional differential equations in the field of viscoelasticity. Therefore, at the end of the paper, this application is presented particularly regarding the relationship between stress and strain for solids and for Newtonian fluids.

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Section: Meanwhile Adam Loverro Used the Fractional Derivative In Phmentioning

confidence: 99%

Fractional Differential Equation as a Models of Newton Fluids for Stress and Strain Problems

Rusyaman¹,

Chaerani²,

Parmikanti³

2018

Proceedings of the 1st Annual International Conference on Mathematics, Science, and Education (ICoMSE 2017)

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“…Untuk menentukan dan menghitung nilai energi potensial minimum pada penelitian sebelumnya, yaitu pada [2] dan [3], harus menggunakan software melalui ratusan bahkan ribuan iterasi sehingga kurang efektif baik dalam sisi proses maupun waktu. Dengan demikian perlu dibuatkan suatu model yang relatif memudahkan perhitungan.…”

Section: Permasalahanunclassified

Model Regresi Energi Potensial Minimum pada Permukaan Hasil Interpolasi

Rusyaman

Carnia

Parmikanti

2014

JMI

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ABSTRAKJika beberapa titik pada sebuah permukaan elastis berbentuk persegi ditekan dari bawah, maka akan terbentuk sebuah permukaan baru yang dapat dinyatakan sebagai fungsi dua variabel hasil interpolasi yang meminimumkan energi.[2], [4], dan [6]. Energi potensial yang diformulasikan sebagai integral dari kuadrat operator Laplace dan diperluas menjadi orde fraksional ini, akan dipengaruhi oleh besarnya tekanan dan elastisitas permukaan. Makalah ini membahas tentang besarnya pengaruh dua variabel bebas yaitu tekanan dan elastisitas terhadap variabel terikat yaitu energi potensial yang terbentuk, serta bagaimana hubungan ketiga variabel tersebut yang dinyatakan dalam bentuk model regresi. Dengan terlebih dahulu mengkarakterisasi orde turunan fraksional menjadi tiga klasifikasi, maka telah dihasilkan tiga buah model untuk tiga keadaan. Dari ketiga model regresi yang dihasilkan menunjukkan bahwa pengaruh bersama variabel elastisitas (orde fraksional) dan variabel besaran tekanan terhadap energi potensial minimum adalah cukup besar, yaitu diatas 85%. Sisanya adalah pengaruh lain yang belum terduga. Kata kunci: energi, elastisitas, fraksional, pemodelan, sinus ganda ABSTRACT If some points on a square-shaped elastic surface is pressed from the bottom, it will form a new surface that can be expressed as a function of two variables interpolated by minimizing the energy.[2], [4], and [6]. Potential energy was formulated as the integral of the square of the Laplace operator and expanded into

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“…The function may be expressed as a double Fourier sine series. As in [3], we also provide an iterative procedure to obtain the function, and explain how it works through an example.…”

Section: > mentioning

confidence: 99%

“…To be concrete, see the example below. For similar ideas in the one dimensional case, we refer the reader to [3].…”

Section: The Procedures To Find the Solutionmentioning

confidence: 99%

Surfaces with Prescribed Nodes and Minimum Energy Integral of Fractional Order

Gunawan¹,

Rusyaman²,

Ambarwati³

2011

itbj.sci.

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his paper presents a method of finding a continuous, real-valued, function of two variables z = u(x,y) defined on the square S := [0,1]^2, which minimizes an energy integral of fractional order, subject to the condition u(0,y) = u(1,y) = u(x,0) = u(x,1) = 0 and u(x_i,y_j) = c_{ij}, where 0 < x_1 < ... < x_M < 1, 0 < y_1 < ... є ℝ are given. The function is expressed as a double Fourier sine series, and an iterative procedure to obtain the function will be presented

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An Interpolation Method That Minimizes an Energy Integral of Fractional Order

Cited by 5 publications

References 8 publications

Fractional Differential Equation as a Models of Newton Fluids for Stress and Strain Problems

Fractional Differential Equation as a Models of Newton Fluids for Stress and Strain Problems

Model Regresi Energi Potensial Minimum pada Permukaan Hasil Interpolasi

Surfaces with Prescribed Nodes and Minimum Energy Integral of Fractional Order

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