The Laplace Transformation (Lectures on applied mathematics volume 1)
The basic inversion formula which works for real time functions assumes the following form [ 34 ]. Zakian developed a method [ 35 ], in which approximation for f t is given by.
Talbot developed a method for the numerical inversion of the Laplace transform, in which the inversion is approximated by the Trapezoidal rule along a special deformed contour. Talbot provided [ 36 ] a recipe for a contour, in which the contour is moved to the left so as to reduce in magnitude the factor e st in the integrand, but the contour must not be moved too close to singularities of F s , as to do so will result in peaks in the integrand function. This requires to be known the locations of the singularities of F s .go to link
Laplace transform 1
Weideman [ 38 ] proposed an improved formula involving the cotangent function with additional parameters. The numerical Inverse Laplace Transform is an ill-posed problem by inherent sensitivity due to the multiplication by an exponential function of time:. To counteract possible issues we apply some modern programming tools and techniques briefly described in the next two subsections. It also has especially clean and straightforward syntax. MPFR supports arbitrary precision floating point variables. It also provides proper rounding of all implemented operations and mathematical functions.
The mpmath [ 41 ] library is a free Python library for real and complex floating-point arithmetic with arbitrary precision. Almost any calculation can be performed just as well at digit or digit precision, with either real or complex numbers, and in many cases they implement efficient algorithms that scale well for extremely high precision work, Proper rounding in compliance with IEEE standard, A high number of special functions, with arbitrary precision and full support for complex numbers, Rudimentary support for interval arithmetic.
The libraries enable a user to set the precision of the arbitrary precision variables by specifying the number of bits to use in the mantissa of the floating point number. Due to the design of the libraries it is possible to work with any precision between 2 bits and maximum bits allowed for a computer.
The most common errors in numerical calculations are caused by wrong rounding. The libraries support proper rounding in compliance with the IEEE standard, as well as an additional one not included in it, i. An increased computational time complexity for arbitrary precision calculations, which is their main disadvantage, is caused not only by applying variables of increased precision more bits for mantissa in the floating point number , but also by applying the corresponding functions and different syntax, which is required.
This simplifies programming and does not increases computational time complexity. For example, computational time complexity using this wrapper for up to digits precision computing does not increase it more than it is in case of double precision application done on mid laptop. We spent a large amount of time studying papers devoted to numerical Inverse Laplace Transform problem and solutions of fractional order differential equations. To be able to select aptly Laplace transforms which are useful in both areas, we consulted numerous noble mathematicians actively involved in FODE about the subject.
As a result, we formed a list with Laplace transforms, which are the most frequently used and which represent various kinds of inversion problems. The list also included transforms which represent generally difficult computational problems. We have considered various accuracy assessment criteria of numerically obtained results which are usually applied in this kind of research. Due to the main goal of the numerical experiment, we selected relative error measure a for accuracy evaluation. It is commonly applied in similar to our comparisons. It actually can also be considered as b , because it informs well how accurate to an exact inversion an approximation at each t is.
However, due to the form of results presentation we selected method a because advantages of b can be utilized for results presented in tables. The criterion c is abandoned due to the irrelevancy to the goals of the experiment. Order of convergence for each method can be found in source papers. They are presented in the references section. The only drawback of b is an increasing value of relative error in case of decreasing values of exact inversion.
It gives a false conclusion about an with time deteriorating accuracy. In such situation there is advised comparison of inversion and relative error plots. The next section presents accuracy comparison results in form of two plots for each selected Laplace transforms: the first plot of an actual approximation by applying a respective numerical inversion method and the second plot with relative error e r of an approximation at each t for each of the methods v c computed in respect to the values of an analytical inversion formula v e.
Citation: Applied Mathematics and Nonlinear Sciences 3, 2; FC is an extension of classical calculus of integer order. This fact enables finding solutions of a problem within the available numerical methods. The solution of the problem selected in the paper focuses on the selection and the programming of seven algorithms of NumILPT. The methods represent four main approaches to the numerical approximation of the Inverse Laplace Transform. They were evaluated against a test set of inversion problems frequently used in FC and representing difficult computational issues.
Due to the general application requirement and clarity of presentation for the present evaluation, remaining methods were left aside. Their performance is either mediocre or they calculated satisfactory accurate inversions in individual cases only. It suggests their application only on an particular inversion problem basis. During the evaluation, several test problems have correctly been inverted by applying more than one algorithm. Here, the Talbot method, which uses the Trapezoidal Rule for the integration of the cotangent function deformed Bromwich contour, is the most accurate and versatile.
However, it requires precise input of Laplace transform poles coordinates for high-accuracy inversions, which can be difficult and can not be automated. For the best results, manual calculation and input of the coordinates is advised. By applying the DeHoog algorithm, which uses Fourier series expansion integration, one can obtain equally accurate inversions.
Wherein, the poles coordinates input is not required. The Abate and Whitt, another algorithm, which employs the use of Fourier series expansion approximation as well, is advised for inverting Laplace transforms, whose originals are the periodic functions. The commonly used standard double precision computer arithmetic has many flaws and limitations, which include the limitations of the values, the double precision variables can hold, no influence of a user on the method of mathematical operations rounding or precision of the intermediate variables.
The decision in these cases is made on an operating systems or compiler level. The state of the art technology called the infinite precision computing [ 45 ] enables eliminating most of these problems by defining of arbitrary precision variables, allowing a user to access the intermediate mathematical operations rounding modes by breaking them down into basic ones and the capability of a user to explicitly choose one of five rounding modes for each mathematical operation.
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Actions Shares. Embeds 0 No embeds. No notes for slide. Amity university sem ii applied mathematics ii lecturer notes 1. Page 14 Type2: 1. The trial solution to be assumed in each case depend on the form of X. Choose PI from the following table depending on the nature of X.
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Solve by the method of undetermined coefficients 2 3 D 3D 2 y 4e x xx c ececy mmmmmSol 2 21 2 2,10 2 1 Assume PI x p ecy 3 1 substituting this in the given d. Page 24 x p xx xxxx x ey cece ececece eyDD 3 33 32 2 4 23 2. Solve x exy dx dy dx yd 2 2 2 by the method of undetermined coefficients. Sol: We have x exyDD 32 42 2 A. E is i i mmm 31 2 2 xcxcey x c 3sin3cos 21 Assume PI in the form x eaaxaxay 2 1 x x eaayD eaaxaDy 41 2 2 2 Substituting these values in the given d. Page 25 3. Solve by using the method of undetermined coefficients xexy dx yd x 3sin9 23 2 2 Sol: We have xexyD x 3sin 9 A.
A differential equation of first order and nth degree is the form 1 2 0 1 This being a differential equation of first order, the associated general solution will contain only one arbitrary constant. We proceed to discuss equations solvable for P or y or x, wherein the problem is reduced to that of solving one or more differential equations of first order and first degree. Equations solvable for p Supposing that the LHS of 1 is expressed as a product of n linear factors, then the equivalent form of 1 is 1 2 1 2 , , They can be solved by the known methods.
Note: We need to present the general solution with the same arbitrary constant in each factor.
The method of solving is illustrated stepwise. By recognizing that the equation is solvable for y, We can proceed to differentiate the same w.
Pierre-Simon Laplace - Wikipedia
We notice that 2 is a differential equation of first order in p and x. We solve the same to obtain the solution in the form. Equations solvable for x We say that the given equation is solvable for x, if it is possible to express x in terms of y and p. The method of solving is identical with that of the earlier one and the same is as follows.
Page 43 Differentiate w. Differentiating 1 w. Page 44 2. Obtain the general solution and the singular solution of the equation 2 4 y px p x Sol: The given equation is solvable for y only. Now, to obtain the singular solution, we differentiate this relation partially w. The general solution now becomes, 2 1 1 2 4 xy x x x Thus 2 4 1 0,x y is the singular solution. Page We can as well substitute for in 1 and prese ie p x c or x p c y p p p x p c Note p x c p x c p nt the solution in the form, 1 1 sin cossin y x c x c x c 4 Obtain the general solution and singular solution of the equation 2 2y px p y.
Sol: The given equation is solvable for x and it can be written as Sol: Dividing throughout by p2 , the equation can be written as 2 2 2 2 2 2 2 2 cot 1 cot. Sol: The given equation is solvable for y only. Page 47 7 Solve 3 2 4 8 0 by solving for x. Hence obtain the associated general and singular solutions 2 0xp py kp a 2: 0, by data 2ie. Page 49 Thus the general solution is a y cx k c Now differentiating partially w. Page 50 4 Solve 2 2 2 ,use the substitution ,.
Analysis of these problems leads to partial derivatives and equations involving them. Later we discuss some methods of solving PDE. Definitions: An equation involving one or more derivatives of a function of two or more variables is called a partial differential equation. The order of a PDE is the order of the highest derivative and the degree of the PDE is the degree of highest order derivative after clearing the equation of fractional powers.
A PDE is said to be linear if it is of first degree in the dependent variable and its partial derivative. In each term of the PDE contains either the dependent variable or one of its partial derivatives, the PDE is said to be homogeneous. Otherwise it is said to be a nonhomogeneous PDE.
Partially differentiating ii , c yx z2 Using this in ii and iii yx z y x z a 2 w w w.
Highlights in the History of the Fourier Transform
Note: As another required partial differential equation. Page 55 Sol: Differentiating z partially w. Sol: Differentiating partially w. Page 58 0ydyxdxor x dy y dx and.
Page 60 Sol: Here we first find z by integration and apply the given conditions to determine the arbitrary functions occurring as constants of integration. The given PDF can be written as yx y z x sinsin Integrating w. Page 65 2. Page 68 Hence the solution of the PDE is given by 2 2 '' '' '' 1 2 3. Page 78 Triple Integrals: The treatment of Triple integrals also known as volume integrals in 3 R is a simple and straight extension of the ideas in respect of double integrals.
Let f x,y,z be continuous and single valued function defined over a region V of space. Let V be divided into sub regions 1 2, Let , , k k kx y z be any arbitrary point w w w. Page 79 within or on the boundary of the sub region kv. Sub region approaches zero the sum 1 has a limit then the limit is denoted by , , V f x y z dv This is called the triple integral of f x,y,z over the region V. Note: When an integration is performed w. If the limits are not constants the integration should be in the order in which dx, dy, dz is given in the integral.
Evaluation of the integral may be performed in any order if all the limits are constants. Page 81 3. Page 83 Triple integral in cylindrical polar coordinates Suppose x,y,z are related to three variables , , R z through the the relation cos , sin , , ,x R y R z zthenR z are called cylindriocal polar coordinates; In this case, , , , , x x x R z x y z y y y J R R z R z z z z R z Hence dxdydz has to be changed to R dR d dz Thus we have , , , , R R f x y z dxdydz R z RdRd dz R is the region in which , , R z vary, as x,y,z vary in R , , cos , sin , R z f R R z Triple integral in spherical polar coordinates Suppose x,y,z are related to three variables , , r through the relations sin cos , sin sin , cosx r y r z r.