@Markiyan Hirnyk 

Thank you for your suggestion. It was my mistake.

@Preben Alsholm 

Thank you Preben. Now it works fine. And the fit is pretty good.

I am sorry for the mistakes I made in the correct definition of the function...

Thanks again 

@Preben Alsholm 

Thank you very much for the code. 

The multiplication is intende as:

.../((exp(x)-1)*(1-exp(-x)))

Maple can indeed find the integral but the fitting procedure stops because it finds complex values 

@Markiyan Hirnyk 

The correct name for the model is Bloch-Gruneisen. Sorry for the mistake

This is the Vector temp

300.00714
298.83669
297.7674
296.70869
295.63479
294.63224
293.53586
292.53867
291.51806
290.41303
289.52155
288.45804
287.44449
286.42436
285.3394
283.38431
281.43135
279.24442
277.25669
275.2398
273.3029
271.32747
269.32248
267.31544
265.31448
263.17422
261.26107
259.24429
257.19845
255.18753
253.13175
251.19641
249.18334
247.26218
245.17674
243.27887
241.2373
239.30844
237.28942
235.27434
233.27775
231.20172
229.22844
227.27221
225.31657
223.20785
221.32365
219.27784
217.19873
215.34418
213.3306
211.20793
209.24846
207.29212
205.29396
203.32154
201.3029
199.34064
197.39214
195.30273
193.32799
191.3099
189.29819
187.32474
185.30149
183.39049
181.31988
179.37801
177.37663
175.37846
173.35183
171.38013
169.3657
167.37789
165.39696
163.45607
162.36534
160.40181
158.38956
157.33904
155.43994
154.3652
152.38356
151.31159
149.36795
147.3296
145.42376
143.38652
141.38662
139.38473
137.45947
136.40464
134.38523
132.37287
130.33119
128.48643
127.42486
126.40449
125.41565
124.38774
122.45044
120.39602
118.40311
116.36564
114.36087
112.38626
110.50996
109.49855
108.35157
106.33843
104.45573
103.44944
102.35522
100.4542
99.42582
98.4723
97.43248
96.46358
95.46081
94.42954
93.4942
92.46588
91.53257
90.45919
89.43843
88.44742
87.3322
85.44337
84.53403
83.46403
82.48198
81.48691
80.47881
79.53736
78.50583
77.42602
76.36378
75.4859
74.48631
73.43704
72.41366
71.37522
70.42429
69.40159
68.46423
67.3745
65.46547
64.3731
62.35591
61.32172
59.38256
57.23494
55.15255
53.29356
51.14391
49.08542
46.92805
44.9383
42.72566
39.46788
35.99256
31.1912
23.77312
19.4463
20.05825
18.95613
17.95726
16.94735
15.94338
14.95177
13.95256
12.9515
11.94896
10.94544
9.94804
8.97655
8.0175
8.03029
7.92914
7.83662
7.73374
7.62835
7.52586
7.43294
7.32679
7.22883
7.12623
7.02995
6.92158
6.81535
6.7244
6.62351
6.52401
6.4236
6.32312
6.22269
6.12345
6.02234

Download temp.txt

This is the vector expres

2.56838
2.56258
2.55755
2.55295
2.54827
2.54402
2.53917
2.53497
2.53055
2.52601
2.52223
2.51776
2.51339
2.50925
2.5047
2.4962
2.48827
2.47909
2.4707
2.46214
2.45389
2.44562
2.43721
2.42887
2.42036
2.41158
2.4034
2.39493
2.3866
2.37803
2.3694
2.36155
2.35293
2.3449
2.3362
2.32815
2.31984
2.31153
2.30315
2.29474
2.28633
2.27769
2.26932
2.26119
2.25322
2.24416
2.23621
2.22762
2.21899
2.21112
2.20257
2.19372
2.18545
2.17715
2.16879
2.16034
2.15179
2.14347
2.13529
2.12634
2.11799
2.10929
2.10072
2.09248
2.08363
2.07545
2.06641
2.05808
2.04955
2.04087
2.03207
2.02358
2.01491
2.00609
1.9976
1.98905
1.98419
1.9759
1.96707
1.96245
1.95397
1.94952
1.94089
1.93609
1.9276
1.91849
1.91019
1.90116
1.89231
1.88343
1.87499
1.87024
1.86129
1.85228
1.84325
1.83509
1.83036
1.82565
1.82128
1.81671
1.80799
1.79853
1.78966
1.78034
1.77121
1.76219
1.75362
1.74901
1.74371
1.73451
1.72575
1.72118
1.71604
1.70717
1.70227
1.69794
1.6931
1.68859
1.68375
1.67893
1.67448
1.66976
1.66534
1.66016
1.65537
1.6507
1.64546
1.63639
1.63211
1.62692
1.62221
1.6176
1.61263
1.60809
1.60314
1.59785
1.59272
1.58845
1.58363
1.5784
1.57349
1.56838
1.56368
1.55871
1.55406
1.5488
1.53945
1.53399
1.52412
1.51896
1.50943
1.49885
1.48864
1.47958
1.46888
1.45897
1.4484
1.43887
1.42839
1.41334
1.39791
1.37822
1.35293
1.34127
1.3423
1.34051
1.33869
1.33696
1.3355
1.33428
1.33331
1.33242
1.33194
1.33152
1.33135
1.33141
1.33151
1.33144
1.33163
1.3316
1.33166
1.33161
1.3317
1.33179
1.33189
1.33191
1.33214
1.33218
1.33231
1.33239
1.33247
1.33258
1.33277
1.33286
1.33299
1.33305
1.33309
1.33256

Download exp_res.txt

@acer

Thank you for finding the syntax error. I fixed it but now the procedure has imaginary values. 

You can overcome that by taking the absolute value but I don't know how much it is justified. 

Anyway the problem of the iteration is not solved since the program stops fitting after the first iteration.

I failed to upload the data points in the formaer post.

I try again

This is the vector temp :

temp.txt

this is the vector expres:

exp_res.txt

@Markiyan Hirnyk 

Thank you all for your kind support.

I still don’t get the iterations but the the fitting-function is getting closer to the data.

I have found manually that  with the two parameters.

theta=55

alpha=0.024

the fit starts to be nice looking.

I am pretty sure that if there is a way to force maple to iterate than I will have a satisfactory match.

 I upload the data for you to check. 

Thanks 

@Preben Alsholm

Thank you guys for your suggestions. It’s much better now, at least I got some numbers from the  fit.

The fit itself is not yet good though. The program stops at the first iteration so that the fit with the experimental data is of very poor quality.

Following your suggestions I have introduced the procedure:

> g := proc (T, alpha, theta)

evalf(rho[0]+alpha*(T-T0)*(T-T0)^5*(int(x^5/(exp(x)-1)(1-exp(-x)), x = 0 .. theta/(T-T0)))/theta^5)

end proc;

And evaluated the fit :

> NonlinearFit(g,temp, expres, parameterranges = [0.001 .. 0.1,  50 .. 300], initialvalues=[0.02,300],  output=[parametervalues, residuals]);

Warning, no iterations performed as initial point satisfies first-order conditions

The fit stops after the first iteration and as a consequence the result is really poor. Is there a way to force Maple to make several iterations?

Thank you very muche for the informations you want to share. It is really appreciated.

Best