Applications of a novel integral transform to partial differential equations

Liang, Xin; Gao, Feng; Yong-liang, Gao; Yang, Xiao‐Jun

doi:10.22436/jnsa.010.02.18

Cited by 22 publications

(23 citation statements)

References 17 publications

(16 reference statements)

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“…(2.10) and (2.11), by contrast, are same to the result presented in [9,16]. Similarly, it remains easy to obtain other properties mentioned above for other ITs, when m and n take other values.…”

Section: λ) (T3) Convolution Property: According To the Definition Osupporting

confidence: 81%

“…In previous literatures [6,9,16,[21][22][23]25], the research on the properties of ITs was mainly focused on their linear property, differential property and convolution property, etc.. An important reason is that these properties play the key roles in solving differential equations. For example, in solving the PDEs, we usually transform them into the ordinary differential equations by using the ITs.…”

Section: Basic Properties Of Glt and Gstmentioning

confidence: 99%

“…Considering the real functions φ 1 (λ) , λ > 0, and φ 2 (λ) , λ > 0, their convolution is defined as [7,16]:…”

Section: The Properties Of Gltmentioning

confidence: 99%

“…Specially, the diffusion equations and wave equations, as fundamental equations in applied mathematics and engineering, have been widely studied [7,13,24,25]. According to the research shown in literatures [6,9,16,[21][22][23], several different ITs, such as Laplace transform (LT), Sumudu transform (ST), ELzaki transform (ET), Laplace-Carson transform (LCT), as well as two new integral transforms (NIT1 and NIT2), were equivalent in obtaining the analytic solutions of the diffusion or wave equations. However, because of the small differences in selection of the kernel functions, they differ in practical applications.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, an interesting result discovered by Eltayeb and Kiliçman [4,8,12] was that LT and ST interchange the image of sin(x) and cos(x). In addition, Liang et al [16] graphically illustrated the duality relations of several ITs.…”

Section: Introductionmentioning

confidence: 99%

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Classifications and duality relations for several integral transforms

Liang¹,

Gao²,

Su³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, we classify several integral transforms into two categories according to the types of their kernel functions and two novel definitions of general integral transforms are suggested. Based on the general integral transforms, some of their basic properties are proved. In addition, the dualities between those two kinds of integral transforms are deducted and discussed in detail. The interesting coupling relations in symmetric form is illustrated graphically. The analysis shows that the classifications are reasonable and the dualities are significant.

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Section: λ) (T3) Convolution Property: According To the Definition Osupporting

confidence: 81%

Section: Basic Properties Of Glt and Gstmentioning

confidence: 99%

“…Considering the real functions φ 1 (λ) , λ > 0, and φ 2 (λ) , λ > 0, their convolution is defined as [7,16]:…”

Section: The Properties Of Gltmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Classifications and duality relations for several integral transforms

Liang¹,

Gao²,

Su³

et al. 2017

J. Nonlinear Sci. Appl.

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Numerical inversion of Laplace transform based on Bernstein operational matrix

Rani

Mishra

Cattani

2018

Math Methods in App Sciences

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This paper provides a technique to investigate the inverse Laplace transform by using orthonormal Bernstein operational matrix of integration. The proposed method is based on replacing the unknown function through a truncated series of Bernstein basis polynomials and the coefficients of the expansion are obtained using the operational matrix of integration. This is an alternative procedure to find the inversion of Laplace transform with few terms of Bernstein polynomials.Numerical tests on various functions have been performed to check the applicability and efficiency of the technique. The root mean square error between exact and numerical results is computed, which shows that the method produces the satisfactory results. A rough upper bound for errors is also estimated. KEYWORDSLaplace transform, numerical inverse Laplace transform, operational matrices, orthonormal Bernstein polynomialswhere c is a positive real number and all the poles of F(s) lie at the left side of the line s = c. Sometimes, inverse Laplace transform becomes complicated when the analytic inverse cannot be found. Hence, to calculate the inverse Laplace transform, numerical techniques are adopted.Recently, numerous different methods have been adopted for analytically and numerically inverting the Laplace transform, which is well suited to different types of functions. Lee and Sheen 5 developed analytic continuation of Bromwich integral and determined the integral by using quadrature rule, Dubner and Abate 6 expressed the function in Fourier cosine series, further it was improved by Durbin 7 that the trapezoidal rule can be applied and the function can be expressed in trigonometric series, Iqbal 8 studied the regularization method, Cuomo et al 9 have described a numerical technique based Math Meth Appl Sci. 2018;41:9231-9243.wileyonlinelibrary.com/journal/mma

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